
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 an equilateral triangle with side length 2, point $P$ is an arbitrary point within the triangle. Connect point $P$ to vertices $A$, $B$, and $C$. From point $P$, draw perpendiculars to each of the three sides of the triangle, meeting these sides at points $D$, $E$, and $F$ respectively. What is the area of the shaded region?

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