
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 adjacent angles to this base are $45^{\circ}$ and $15^{\circ}$. From the vertex opposite the base, a circle is drawn with a radius equal to the height dropped to this base. Find the area of the part of the corresponding circle that is enclosed within the triangle.

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