
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
Given a triangle \(ABC\) such that \(\angle ACB = 50^\circ\). The altitude corresponding to vertex \(A\) and the bisector of \(\angle ABC\) intersect at \(P\) within the interior of triangle \(ABC\) and \(\angle APB = 105^\circ\). Find the measures of the angles \(\angle BAC\) and \(\angle ABC\).

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