
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  $\omega$  be a unit circle with center  $O$  and diameter  $AB$ . A point  $C$  is chosen on  $\omega$ . Let  $M$ ,  $N$  be the midpoints of arc  $AC$ ,  $BC$ , respectively, and let  $AN,BM$  intersect at  $I$ . Suppose that  $AM,BC,OI$  concur at a point. Find the area of  $\triangle ABC$ .

*Proposed by Kevin You*

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