
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 vectors $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OB}}$ with an angle of $120^{\circ}$ between them, and with magnitudes $|\overrightarrow{\mathrm{OA}}| = 2$, $|\overrightarrow{\mathrm{OB}}| = 6$. Construct a line segment $\mathrm{PQ}$ with length $2|\overrightarrow{\mathrm{AB}}|$ passing through point $0$, such that points $\mathrm{P}$ and $\mathrm{Q}$ are symmetric with respect to point $0$. Find the angle between vector $\overrightarrow{\mathrm{PQ}}$ and $\overrightarrow{\mathrm{AB}}$ that maximizes $\overrightarrow{\mathrm{AP}} \cdot \overrightarrow{\mathrm{BQ}}$, and determine this maximum value.

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