
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
In parallelogram \(ABCD\), diagonal \(AC\) is perpendicular to side \(AB\). A certain circle touches side \(BC\) of the parallelogram \(ABCD\) at point \(P\) and touches the line passing through vertices \(A\) and \(B\) of the same parallelogram at point \(A\). A perpendicular \(PQ\) is drawn from point \(P\) to side \(AB\) (point \(Q\) is the foot of this perpendicular). Find the angle \(ABC\) if it is known that the area of the parallelogram \(ABCD\) is \(\frac{1}{2}\), and the area of the pentagon \(QPCDA\) is \(S\).

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