
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
A segment of the graph of $y = H(x)$ is shown below. The distance between the grid lines is $1$ unit. Compute $H(H(H(H(H(2)))))$.

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
rr_cartesian_axes(-5,6,-4,8);
real f(real x) {return ((x-2)*(x-2)/2 - 1);}
draw(graph(f,-1.5,5.5,operator ..), blue);
[/asy]

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