
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 the foot \(P\) of the perpendicular from point \(C\) to the line \(AB\) lies within the segment \(AB\). From point \(P\), perpendiculars \(p\) and \(q\) are drawn to the lines \(AC\) and \(BC\) respectively (in the given order). Let \(S\) be the intersection of the line \(BC\) and the line \(q\), and \(T\) be the intersection of the line \(AC\) and the line \(p\). Calculate the angle \(\angle ACB\), given that \(|\angle APT| + |\angle BPS| = 20^\circ\).

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