
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 the right triangle \( ABC \) with a right angle at \( C \), the length of the angle bisector of angle \( A \) is 4, and angle \( A = 60^{\circ} \). On the perpendicular bisector of leg \( CB \) at point \( Q \) lies the center of a circle that touches lines \( AC \) and \( AB \) at points \( K \) and \( M \) respectively. Find the area of triangle \( OQK \), where point \( O \) is the center of the incircle of the triangle \( ABC \).

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