
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

A circle inscribed in an isosceles triangle $ABC$ touches the base $AC$ at point $D$ and the lateral side $AB$ at point $E$. Point $F$ is the midpoint of side $AB$, and point $G$ is the intersection point of the circle and the segment $FD$ other than $D$. The tangent to the circle through point $G$ intersects side $AB$ at point $H$. Find the angle $BCA$, given that $FH:HE = 2:3$.

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