
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
Triangle  $ ABC$  obeys  $ AB = 2AC$  and  $ \angle{BAC} = 120^{\circ}.$  Points  $ P$  and  $ Q$  lie on segment  $ BC$  such that
\begin{eqnarray*}
AB^2 + BC \cdot CP = BC^2 
3AC^2 + 2BC \cdot CQ = BC^2
\end{eqnarray*}
Find  $ \angle{PAQ}$  in degrees.

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