Nutmeg

NAME

nutmeg spice post-processor

SYNOPSIS

nutmeg [ - ] [ -n ] [ -t term ] [ datafile ... ]

DESCRIPTION

Nutmeg is a post processor for SPICE it takes the raw output file created by spice -r and plots the data on a graphics terminal or a workstation display. Note that the raw output file is different from the data that SPICE writes to the standard output.

Arguments are:

 

- Don't try to load the default data file ("rawspice") if no other files are given.

 

-n (or -N)
Don't try to source the file ".spiceinit" upon startup. Normally nutmeg tries to find the file in the current directory, and if it is not found then in the user's home directory.

 

-t term (or -T term)
The program is being run on a terminal with mfb name term.

Further arguments are taken to be data files in binary or ascii format (see sconvert(1)) which are loaded into nutmeg. If the file is in binary format, it may be only partially completed (useful for examining SPICE ouput before the simulation is finished). One file may contain any number of data sets from different analyses.

Nutmeg data is in the form of vectors: time, voltage, etc. Each vector has a type, and vectors can be operated on and combined algebraicly in ways consistent with their types. Vectors are normally created when a data file is read in (see the load command below), and when the initial datafile is loaded. They can also be created with the let command.

An expression is an algebraic formula involving vectors and scalars (a scalar is a vector of length 1), and the following operations:

 

+, -, *, %, /, ^, and ,.

% is the modulo operator, and the comma operator has two meanings: if it is present in the argument list of a user-definable function, it serves to seperate the arguments. Otherwise, the term x , y is synonymous with x + j(y).

Also available are the logical operations & (and), | (or), ! (not), and the relational operations <, , =, <=, =, and < (not equal). If used in an algebraic expression they work like they would in C, producing values of 0 or 1. The relational operators have the following synonyms: "gt" is , "lt" is <, "ge" is =, "le" is <=, "ne" is <, "eq" is =, "and" is &, "or" is |, and "not" is !. These are useful when < and might be confused with IO redirection (which is almost always).

The following functions are available:

 

mag(vector) The magnitude of vector.

ph(vector) The phase of vector.

j(vector) i (sqrt(-1)) times vector.

real(vector) The real component of vector.

imag(vector) The imaginary part of vector.

db(vector) 20 * log10(mag(vector)).

log(vector) The logarithm (base 10) of the vector.

ln(vector) The natural logarithm (base e) of vector.

exp(vector) e to the vector power.

abs(vector) The absolute value of vector.

sqrt(vector) The square root of vector.

sin(vector) The sin of vector.

cos(vector) The cosine of vector.

tan(vector) The tangent of vector.

atan(vector) The inverse tangent of vector.

norm(vector) The vector normalized to 1 (i.e, the largest magnitude of any component will be 1).

rnd(vector) A vector with each component a random integer between 0 and the absolute value of the vectors's corresponding component.

mean(vector) The result is a scalar (a length 1 vector) that is the mean of the elements of vector.

vector(number) The result is a vector of length number, with elements 0, 1, ... number - 1. If number is a vector then just the first element is taken, and if it isn't an integer then the floor of the magnitude is used.

length(vector) The length of vector.

interpolate(plot.vector) The result of interpolating the named vector onto the scale of the current plot. This function uses the variable polydegree to determine the degree of interpolation.

A vector may be either the name of a vector already defined, a floating- point number (a scalar), or a list like [elt1 elt2 ... eltn], which is a vector of length n. A number may be written in any format acceptable to SPICE, such as 14.6MEG or -1.231E-4. Note that you can either use scientific notation or one of the abbreviations like MEG or G, but not both. As with SPICE, a number may have trailing alphabetic characters after it.

The notation expr [lower upper], where lower and upper are numbers, denotes the range of elements from expr between lower and upper. The notation expr [num] denotes the num'th element of expr. If upper is lower than lower, the order of the elements in the vector is reversed. In all other cases, [ and ] serve to surround literal vectors as described above. (You may have to use a lot of parentheses to make sure that you get what you want. For instance, you have to type print (foo) ([1 2]) to print the two vectors. Otherwise it will be interpreted as a function call or a vector with an index.) Note that the expression foo[10 20][5] will not yield the 15th element of foo, but rather the 5th. In general only the last index suffix on an expression will take effect.

To reference vectors in a plot that is not the current plot (see the setplot command, below), the notation plotname.vecname can be used.

Either a plotname or a vector name may be the wildcard all. If the plotname is all, matching vectors from all plots are specified, and if the vector name is all, all vectors in the specified plots are referenced. Note that you may not use binary operations on expressions involving wildcards it is not obvious what all + all should denote, for instance.

Thus some (contrived) examples of expressions are:

cos(TIME) + db(v(3))

sin(cos(log([1 2 3 4 5 6 7 8 9 10])))

TIME * rnd(v(9)) - 15 * cos(vin#branch) ^ [7.9e5 8]

not ((ac3.FREQ[32] & tran1.TIME[10]) gt 3)

Nutmeg commands are as follows:

 

plot exprs [ylimit ylo yhi] [xlimit xlo xhi] [xindices xilo xihi] [xcompress comp] [xdelta xdel] [ydelta ydel] [xlog] [ylog] [vs xname] [xlabel word] [ylabel word] [title word] [samep]

Plot the given exprs on the screen (if you are on a graphics terminal). The xlimit and ylimit arguments determine the high and low x- and y-limits of the axes, respectively. The xindices arguments determine what range of points are to be plotted everything between the xilo'th point and the xihi'th point is plotted. The xcompress argument specifies that only one out of every comp points should be plotted. If an xdelta or a ydelta parameter is present, it specifies the spacing between grid lines on the X- and Y-axis. These parameter names may be abbreviated to xl, yl, xind, xcomp, xdel and ydel respectively. The xname argument is an expression to use as the scale on the x-axis. If xlog or ylog are present, the X or Y scale respectively will be logarithmic. The xlabel and ylabel arguments cause the specified labels to be used for the X and Y axes, respectively. If samep is given, the values of the other parameters (other than xname) from the previous plot, hardcopy, or asciiplot command will be used unless re-defined on the command line. Finally, the title argument will be used in the place of the plot name at the bottom of the graph.

hardcopy file plotargs

Just like plot, except creates a file called file containing the plot. The file is an image in plot(5) format, and can be printed by either the plot(1) program or lpr with the -g flag.

asciiplot plotargs

Produce a line printer plot of the vectors. The plot is sent to the standard output, so you can put it into a file with asciiplot args ... file. The set options width, height, and nobreak determine the width and height of the plot, and whether there are page breaks, respectively. Note that you will have problems if you try to asciiplot something with an X-scale that isn't monotonic (i.e, something like sin(TIME) ), because asciiplot uses a simple-minded sort of linear interpolation.

define function(arg1, arg2, ...) expression

Define the user-definable function with the name function and arguments arg1, arg2, ... to be expression, which may involve the arguments. When the function is later used, the arguments it is given are substituted for the formal arguments when it is parsed. If expression is not present, any definition for function is printed, and if there are no arguments to define then all currently active definitions are printed. Note that you may have different functions defined with the same name but different arities. Some useful definitions are:

define max(x,y) (x y) * x + (x <= y) * y

define min(x,y) (x < y) * x + (x = y) * y

undefine function ...

Definitions for the named user-defined functions are deleted.

let name = expr

Creates a new vector called name with the value specified by expr, an expression as described above. If expr is [] (a zero-length vector) then the vector becomes undefined. If there are no arguments, let is the same as display.

print [col] [line] expr ...

Prints the vector described by the expression expr. If the col argument is present, print the vectors named side by side. If line is given, the vectors are printed horizontally. col is the default, unless all the vectors named have a length of one, in which case line is the default. The options width, length, and nobreak are effective for this command (see asciiplot). If the expression is all, all of the vectors available are printed. Thus print col all file will print everything in the file in SPICE2 format. The scale vector (time, frequency) will always be in the first column unless the variable noprintscale is true.

load [filename] ...

Loads the raw data in either binary or ascii format from the files named. The default filename is rawspice, or the argument to the -r flag if there was one.

source filename

Reads commands from the file filename. Lines beginning with the character * are considered comments and ignored.

help [all] [command ...]

Prints help. If the argument all is given, a short description of everything you could possibly type is printed. If commands are given, descriptions of those commands are printed. Otherwise help for only a few major commands is printed.

display [varname ...]

Prints a summary of currently defined vectors, or of the names specified. The vectors are sorted by name unless the variable nosort is set. The information given is the name of the vector, the length, the type of the vector, and whether it is real or complex data. Additionally, one vector will be labeled [scale]. When a command such as plot is given without a vs argument, this scale is used for the X-axis. It is always the first vector in a rawfile, or the first vector defined in a new plot. If you undefine the scale (i.e, let TIME = []), a random remaining vector will become the scale.

setplot [plotname]

Set the current plot to the plot with the given name, or if no name is given, prompt the user with a menu. (Note that the plots are named as they are loaded, with names like tran1 or op2. These names are shown by the setplot and display commands and are used by diff, below.) If the "New plot" item is selected, the current plot will become one with no vectors defined. Note that here the word "plot" refers to a group of vectors that are the result of one SPICE run. When more than one file is loaded in, or more than one plot is present in one file, nutmeg keeps them seperate and only shows you the vectors in the current plot.

settype type vector ...

Change the type of the named vectors to type. Type names can be found in the manual page for sconvert.

diff plot1 plot2 [vec ...]

Compare all the vectors in the specified plots, or only the named vectors if any are given. There are different vectors in the two plots, or any values in the vectors differ significantly the difference is reported. The variables abstol, reltol, and vntol are used to determine what "significantly" means (see the SPICE3 User's Manual).

quit

Quit nutmeg

bug

Send a bug report. (If you have defined BUGADDR, the mail will go there.)

write [file] [exprs]

Writes out the expr's to file. First vectors are grouped together by plots, and written out as such. (I.e, if the expression list contained three vectors from one plot and two from another, then two plots will be written, one with three vectors and one with two.) Additionally, if the scale for a vector isn't present, it is automatically written out as well. The default format is ascii, but this can be changed with the set filetype command. The default filename is rawspice, or the argument to the -r flag on the command line, if there was one, and the default expression list is all.

shell [args ...]

Fork a shell, or execute the arguments as a command to the shell.

alias [word] [text ...]

Causes word to be aliased to text. History substitutions may be used, as in C-shell aliases.

unalias [word ...]

Removes any aliases present for the words.

history [number]

Print out the history, or the last number commands typed at the keyboard. Note: in \\*S version 3a7 and earlier, all commands (including ones read from files) were saved.

set [word] [word = value] ...

Set the value of word to be value, if it is present. You can set any word to be any value, numeric or string. If no value is given then the value is the boolean 'true'. The value of word may be inserted into a command by writing $word. If a variable is set to a list of values that are enclosed in parentheses (which must be seperated from their values by white space), the value of the variable is the list. The variables meaningful to nutmeg (of which there are too many) are:

abstol

The absolute tolerance used by the diff command.

appendwrite

Append to the file when a write command is issued, if one already exists.

colorN

These variables determine the colors used, if X is being run on a color display. N may be between 0 and 15. Color 0 is the background, color 1 is the grid and text color, and colors 2 through 15 are used in order for vectors plotted. The value of the color variables should be names of colors, which may be found in the file /usr/lib/rgb.txt.

combplot

Plot vectors by drawing a vertical line from each point to the X-axis, as opposed to joining the points. Note that this option is subsumed in the plottype option, below.

cpdebug

Print cshpar debugging information. (Must be complied with the -DCPDEBUG flag.)

debug

If set then a lot of debugging information is printed. (Must be compiled with the -DFTEDEBUG flag.)

device

The name (/dev/tty??) of the graphics device. If this variable isn't set then the user's terminal is used. To do plotting on another monitor you will probably have to set both the device and term variables. (If device is set to the name of a file, nutmeg will dump the graphics control codes into this file -- this is useful for saving plots.)

echo

Print out each command before it is executed.

filetype

This can be either ascii or binary, and determines what the format of rawfiles will be. The default is ascii.

fourgridsize

How many points to use for interpolating into when doing fourier analysis.

gridsize

If this variable is set to an integer, this number will be used as the number of equally spaced points to use for the Y-axis when plotting. Otherwise the current scale will be used (which may not have equally spaced points). If the current scale isn't strictly monotonic, then this option will have no effect.

hcopydev

If this is set, when the hardcopy command is run the resulting file is automatically printed on the printer named hcopydev with the command lpr -Phcopydev -g file.

hcopydevtype

This variable specifies the type of the printer output to use in the hardcopy command. If hcopydevtype is not set, plot (5) format is assumed. The standard distribution currently recognizes postscript as an alternative output format. When used in conjunction with hcopydev, hcopydevtype should specify a format supported by the printer.

height

The length of the page for asciiplot and print col.

history

The number of events to save in the history list.

nfreqs

The number of frequencies to compute in the fourier command. (Defaults to 10.)

nobreak

Don't have asciiplot and print col break between pages.

noasciiplotvalue

Don't print the first vector plotted to the left when doing an asciiplot.

noclobber

Don't overwrite existing files when doing IO redirection.

noglob

Don't expand the global characters `*', `?', `[', and `]'. This is the default.

nogrid

Don't plot a grid when graphing curves (but do label the axes).

nomoremode

If nomoremode is not set, whenever a large amount of data is being printed to the screen (e.g, the print or asciiplot commands), the output will be stopped every screenful and will continue when a carriage return is typed. If nomoremode is set then data will scroll off the screen without hesitation.

nonomatch

If noglob is unset and a global expression cannot be matched, use the global characters literally instead of complaining.

nosort

Don't have display sort the variable names.

noprintscale Don't print the scale in the leftmost column when a print col command is given.

numdgt

The number of digits to print when printing tables of data (fourier, print col). The default precision is 6 digits. On the VAX, approximately 16 decimal digits are available using double precision, so numdgt should not be more than 16. If the number is negative, one fewer digit is printed to ensure constant widths in tables.

plottype

This should be one of normal, comb, or point:chars. normal, the default, causes points to be plotted as parts of connected lines. comb causes a comb plot to be done (see the description of the combplot variable above). point causes each point to be plotted seperately the chars are a list of characters that will be used for each vector plotted. If they are omitted then a default set is used.

polydegree

The degree of the polynomial that the plot command should fit to the data. If polydegree is N, then nutmeg will fit a degree N polynomial to every set of N points and draw 10 intermediate points in between each endpoint. If the points aren't monotonic, then it will try rotating the curve and reducing the degree until a fit is achieved.

polysteps The number of points to interpolate between every pair of points available when doing curve fitting. The default is 10. (This should really be done automatically.)

program The name of the current program (argv[0]).

prompt The prompt, with the character `!' replaced by the current event number.

rawfile The default name for rawfiles created.

reltol The relative tolerance used by the diff command.

rhost The machine to use for remote SPICE-3 runs, instead of the default one. (See the description of the rspice command, below.)

rprogram The name of the remote program to use in the rspice command.

slowplot Stop between each graph plotted and wait for the user to type return before continuing.

sourcepath A list of the directories to search when a source command is given. The default is the current directory and the standard spice library (/usr/local/lib/spice, or whatever LIBPATH is #defined to in the \\*S source.

spicepath The program to use for the aspice command. The default is /cad/bin/spice.

term The mfb name of the current terminal.

units If this is degrees, then all the trig functions will use degrees instead of radians.

unixcom If a command isn't defined, try to execute it as a UNIX command. Setting this option has the effect of giving a rehash command, below. This is useful for people who want to use nutmeg as a login shell.

verbose Be verbose. This is midway between echo and debug / cpdebug.

vnto The absolute voltage tolerance used by the diff command.

width The width of the page for asciiplot and print col.

xbrushheight The height of the brush to use if X is being run.

xbrushwidth The width of the brush to use if X is being run.

xfont The name of the X font to use when plotting data and entering labels. The plot may not look entirely great if this is a variable-width font.

unset [word] ... Unset the variables word.

shift [varname] [number] If varname is the name of a list variable, it is shifted to the left by number elements. (I.e, the number leftmost elements are removed.) The default varname is argv, and the default number is 1.

rusage [resource ...] Print resource usage statistics. If any resources are given, just print the usage of that resource. Currently valid resources are:

elapsed The amount of time elapsed since the last rusage elaped call.

faults Number of page faults and context switches (BSD only).

space Data space used.

time CPU time used so far.

everything All of the above.

cd [directory] Change the current working directory to directory, or to the user's home directory if none is given.

aspice [output-file] Start a SPICE-3 run, and when it is finished load the data. The raw data is kept in a temporary file. If output-file is specified then the diagnostic output is directed into that file, otherwise it is thrown away.

jobs Report on the asynchronous SPICE-3 jobs currently running. Nutmeg checks to see if the jobs are finished every time you execute a command. If it is done then the data is loaded and becomes available.

rspice [input file]

Runs a SPICE-3 remotely taking the input file as a SPICE-3 input deck, or the current circuit if no argument is given. Nutmeg waits for the job to complete, and passes output from the remote job to the user's standard output. When the job is finished the data is loaded in as with aspice. If the variable rhost is set, nutmeg will connect to this host instead of the default remote SPICE-3 server machine. Note that this command will only work if your system administrator is running a SPICE-3 daemon on the remote host. If the variable rprogram is set, then rspice will use this as the pathname to the program to run.

echo [stuff...] Echos the arguments.

fourier fundamental_frequency [value ...]

Does a fourier analysis of each of the given values, using the first 10 multiples of the fundamental frequency (or the first nfreqs, if that variable is set see below). The output is like that of the .four \\*S card. The values may be any valid expression. The values are interpolated onto a fixed-space grid with the number of points given by the fourgridsize variable, or 200 if it is not set. The interpolation will be of degree polydegree if that variable is set, or 1. If polydegree is 0, then no interpolation will be done. This is likely to give erroneous results if the time scale is not monotonic, though.

version [version id] Print out the version of nutmeg that is running. If there are arguments, it checks to make sure that the arguments match the current version of SPICE. (This is mainly used as a Command: line in rawfiles.)

rehash Recalculate the internal hash tables used when looking up UNIX commands, and make all UNIX commands in the user's PATH available for command completion. This is useless unless you have set unixcom first (see above).

The following control structures are available:

while condition

statement

...

end

While condition, an arbitrary algebraic expression, is true, execute the statements.

repeat [number]

statement

...

end

Execute the statements number times, or forever if no argument is given.

dowhile condition

statement

...

end

The same as while, except that the condition is tested after the statements are executed.

foreach var value ...

statement

...

end

The statements are executed once for each of the values, each time with the variable var set to the current one. (var can be accessed by the $var notation see below).

if condition

statement

...

else

statement

...

end

If the condition is non-zero then the first set of statements are executed, otherwise the second set. The else and the second set of statements may be omitted.

label word

If a statement of the form goto word is encountered, control is transfered to this point, otherwise this is a no-op.

goto word

If a statement of the form label word is present in the block or an enclosing block, control is transfered there. Note that if the label is at the top level, it must be before the goto statement (i.e, a forward goto may occur only within a block).

continue

If there is a while, dowhile, or foreach block enclosing this statement, control passes to the test, or in the case of foreach, the next value is taken. Otherwise an error results.

break

If there is a while, dowhile, or foreach block enclosing this statement, control passes out of the block. Otherwise an error results.

Of course, control structures may be nested. When a block is entered and the input is the terminal, the prompt becomes a number of 's equalling the number of blocks the user has entered. The current control structures may be examined with the debugging command cdump.

If a word is typed as a command, and there is no built-in command with that name, the directories in the sourcepath list are searched in order for the file. If it is found, it is read in as a command file (as if it were sourced). Before it is read, however, the variables argc and argv are set to the number of words following the filename on the command line, and a list of those words respectively. After the file is finished, these variables are unset. Note that if a command file calls another, it must save its argv and argc since they will get altered. Also, command files may not be re-entrant since there are no local variables. (Of course, the procedures may explicitly manipulate a stack...) This way one can write scripts analogous to shell scripts for nutmeg and \\*S. Note that for the script to work with \\*S, it must begin with a blank line (or whatever you like, since it will be thrown away) and then a line with .control on it. This is an unfortunate result of the source command being used for both circuit input and command file execution. Note also that this allows the user to merely type the name of a circuit file as a command, and it will be automatically run.

There are various command scripts installed in /usr/local/lib/spice/scripts (or whatever the path is on your machine), and the default sourcepath includes this directory, so you can use these command files (almost) like builtin commands.

Nutmeg will use either X or MFB, depending on whether it finds the variable DISPLAY in the environment. If you are using X on a workstation, it should already be present, but if you want to display graphics on a different machine than the one you are running nutmeg on, DISPLAY should be of the form machine:0.

If X is being used, the cursor may be positioned at any point on the screen when the window is up and characters typed at the keyboard will be added to the window at that point. The window may then be sent to a printer using the xpr(1) program.

There are a number of pre-defined constants in nutmeg. They are:

pi

e

The base of natural logarithms

c

The speed of light

i

The square root of -1

kelvin

Absolute 0 in Centigrade

echarge

The charge on an electron

boltz

Boltzman's constant

planck

Planck's constant (h)

These are all in MKS units. If you have another variable with a name that conflicts with one of these then it takes precedence.

Nutmeg occasionally checks to see if it is getting close to running out of space, and warns the user if this is the case. (This is more likely to be useful with the SPICE front end.)

C-shell type quoting with "" and '', and backquote substitution may be used. Within single quotes, no further substitution (like history substitution) is done, and within double quotes, the words are kept together but further substitution is done. Any text between backquotes is replaced by the result of executing the text as a command to the shell.

Tenex-style ('set filec' in the 4.3 C-shell) command, filename, and keyword completion is possible: If EOF (control-D) is typed after the first character on the line, a list of the commands or possible arguments is printed. (If it is alone on the line it will exit nutmeg.) If escape is typed, then nutmeg will try to complete what the user has already typed. To get a list of all commands, the user should type &ltspace ^D.

The values of variables may be used in commands by writing $varname where the value of the variable is to appear. The special variables $$ and $< refer to the process ID of the program and a line of input which is read from the terminal when the variable is evaluated, respectively. If a variable has a name of the form $&word, then word is considered a vector (see above), and its value is taken to be the value of the variable. If $foo is a valid variable, and is of type list, then the expression $foo[low-high] represents a range of elements. Either the upper index or the lower may be left out, and the reverse of a list may be obtained with $foo[len-0]. Also, the notation $?foo evaluates to 1 if the variable foo is defined, 0 otherwise, and $#foo evaluates to the number of elements in foo if it is a list, 1 if it is a number or string, and 0 if it is a boolean variable.

History substitutions, similar to C-shell history substitutions, are also available see the C-shell manual page for all of the details.

The characters ~, {, and } have the same effects as they do in the C-Shell, i.e., home directory and alternative expansion. It is possible to use the wildcard characters *, ?, [, and ] also, but only if you unset noglob first. This makes them rather useless for typing algebraic expressions, so you should set noglob again after you are done with wildcard expansion. Note that the pattern [^abc] will match all characters except a, b, and c.

IO redirection is available the symbols , , &, &, and < have the same effects as in the C-shell.

You may type multiple commands on one line, seperated by semicolons.

If you want to use a different mfbcap file than the default (usually ~cad/lib/mfbcap), you have to set the environment variable MFBCAP before you start nutmeg. The -m option and the mfbcap variable no longer work.

 

BUGS

The label entry facilities are very primitive after all, nutmeg isn't a graphics editor (yet). You must be careful to type very slowly when entering labels -- nutmeg checks the X event queue once every second, and can get very confused if characters arrive faster than that.

If you redefine colors after creating a plot window with X, and then cause the window to be redrawn, it will not to the right thing.

When defining aliases like

alias pdb plot db( '!:1' - '!:2' )

you must be careful to quote the argument list substitutions in this manner. If you quote the whole argument it might not work properly.

In a user-defined function, the arguments cannot be part of a name that uses the plot.vec syntax. I.e,

define poke(duck) cos(tran1.duck)

won't do the right thing.

If you type plot all all, or otherwise use a wildcard reference for one plot twice in a command, bad things will happen.

The asciiplot command doesn't deal with log scales or the delta keywords.

There are probably some features that nutmeg doesn't have yet.

CAVEATS

Often the names of terminals recognised by MFB are different from those in /etc/termcap. Thus you may have to reset your terminal type with the command

set term = termname

where termname is the name in the mfbcap file.

The hardcopy command is useless on VMS and other systems without the plot command, unless the user has a program that understands plot(5) format.

VMS NOTES

Nutmeg can be run under VAX/VMS. Some features like command, etc completion, expansion of *, ?, and [], backquote substitution, the shell command, and so forth do not work. (In fact command completion only works on 4.2 or 4.3 BSD.)

Nutmeg will look for start-up commands in the file spice.rc in the current directory.

The standard suffix for rawspice files in VMS is ".raw".

You will have to respond to the -more- prompt during plot with a carriage return instead of any key as you can do on UNIX.

 

 

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