
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
Let \(ABC\) be an acute-angled triangle with \(AB \neq BC\) and circumcircle \(k\). Let \(P\) and \(Q\) be the intersections of \(k\) with the angle bisector and the external angle bisector of \(\angle CBA\) respectively. Let \(D\) be the intersection of \(AC\) and \(PQ\). Determine the ratio \(AD:DC\).

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