
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
Let $P$ be a point inside triangle $ABC$. Let $I_1$, $I_2$, and $I_3$ be the incenters of triangles $PBC$, $PCA$, and $PAB$, respectively. If the area of triangle $ABC$ is $36$, then find the area of triangle $I_1 I_2 I_3$.

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