
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
An equilateral pentagon  $AMNPQ$  is inscribed in triangle  $ABC$  such that  $M\in\overline{AB}$ ,  $Q\in\overline{AC}$ , and  $N,P\in\overline{BC}$ .
  
Suppose that  $ABC$  is an equilateral triangle of side length  $2$ , and that  $AMNPQ$  has a line of symmetry perpendicular to  $BC$ . Then the area of  $AMNPQ$  is  $n-p\sqrt{q}$ , where  $n, p, q$  are positive integers and  $q$  is not divisible by the square of a prime. Compute  $100n+10p+q$ .

*Proposed by Michael Ren*

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