
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
The base of a triangle is $a$, and the angles adjacent to it measure $45^{\circ}$ and $15^{\circ}$. A circle with a radius equal to the altitude dropped to this base is drawn from the vertex opposite the base. Find the area of the part of the corresponding circle that is contained within the triangle.

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