
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 \( ABC \) has \( AB = 2AC \) and \( \angle BAC = 120^\circ \). Points \( P \) and \( Q \) lie on segment \( BC \) such that 
\[ 
\begin{aligned}
AB^2 + BC \cdot CP &= BC^2, \\
3AC^2 + 2BC \cdot CQ &= BC^2.
\end{aligned}
\]
Find \(\angle PAQ\) in degrees.

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