
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 the incircle of  $\triangle ABC$  be tangent to  $AB,BC,AC$  at points  $M,N,P$ , respectively. If  $\measuredangle BAC=30^\circ$ , find  $\tfrac{[BPC]}{[ABC]}\cdot\tfrac{[BMC]}{[ABC]}$ , where  $[ABC]$  denotes the area of  $\triangle ABC$ .

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