
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 an acute-angled triangle \(ABC\) with side \(AC\) equal to \(b\), \(\angle A = \alpha\), and \(\angle B = \beta\) radians, a circle is drawn through the orthocenter (the point of intersection of the altitudes) and the feet of the altitudes dropped to sides \(AB\) and \(BC\). Find the area of the common part of the triangle and the circle.

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