
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
Given the function \( f(x) \) defined only for \( x \) values from -5 to 5 as shown in the graph below, determine how many values of \( x \) satisfy \( f(f(x)) = 0 \).

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

real lsf=0.5;
pen dps=linewidth(0.7)+fontsize(10);
defaultpen(dps); pen ds=black;

real xmin=-5.5,xmax=5.5,ymin=-1.5,ymax=5.5;

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 -0.5*x^2 + x + 3; }
draw(graph(f,-5,-1));
draw((-1,3.5)--(1,1.5));
real g(real x) { return 0.5*x^2 - x + 1.5; }
draw(graph(g,1,5));

dot("$(-5,3)$", (-5,3), NW);
dot("$(-1,3.5)$", (-1,3.5), N);
dot("$(0,3)$", (0,3), NE);
dot("$(1,1.5)$", (1,1.5), S);
dot("$(3,3)$", (3,3), SE);
dot("$(5,5)$", (5,5), NE);

label("$f(x)$", (-4,5), E);
[/asy]

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