
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
In \(\triangle ABC\), \(\omega\) is the circumcircle, \(I\) is the incenter, and \(I_A\) is the \(A\)-excenter. Let \(M\) be the midpoint of arc \(\widehat{BAC}\) on \(\omega\), and suppose that \(X\) and \(Y\) are the projections of \(I\) onto \(MI_A\) and \(I_A\) onto \(MI\), respectively. If \(\triangle X Y I_A\) is an equilateral triangle with a side length of 1, compute the area of \(\triangle ABC\).

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