
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
The graph of \( f(x) \) is given below, where \( f \) is defined for \( x \) between \(-4\) and \( 4\) inclusive. Determine the number of \( x \) values that satisfy \( f(f(x)) = 1 \).

[asy]
import graph; size(9cm);

real lsf=0.5;

pen dps=linewidth(0.7)+fontsize(10);

defaultpen(dps); pen ds=black;

real xmin=-4.5,xmax=4.5,ymin=-0.5,ymax=4.5;

Label laxis; laxis.p=fontsize(10);

xaxis("$x$",xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true);
yaxis("$y$",ymin,ymax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true);

real f(real x) { return -.5*x^2-1.5*x+2;}
draw(graph(f,-4,-1));
draw((-1,1.5)--(1,1));
draw((1,0)--(3.5,3.5));
real f(real x) { return 1.5*x-1.5;}
draw(graph(f,3.5,4));

dot("$(-4,0)$", (-4,0), NW);
dot("$(-3,2)$", (-3,2), NW);
dot("$(-1,1.5)$", (-1,1.5), N);
dot("$(-1,1)$", (-1,1), N);
dot("$(1,1)$", (1,1), S);
dot("$(4,4)$", (4,4), NE);

label("$f(x)$",(-3,3.5),E);

[/asy]

Present the answer in LaTex format: \boxed{Your answer}
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