
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 triangle $ABC$ be given with point $Q$ in the same plane. Point $Q$ is equidistant from $B$ and $C$, angle $BQC$ is thrice angle $BAC$, and line $\overline{AB}$ intersects $\overline{CQ}$ at point $E$. Given $CQ = 4$ and $EQ = 3$, calculate the product $BE \cdot EC$.

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