
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
Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP = 3$ and $HQ = 4$, calculate $(BP)(PC) - (AQ)(QC)$. Assume $\triangle BPH \sim \triangle APC$ and $\triangle AQH \sim \triangle BQC$, and the ratio of the side lengths split by the altitudes are maintained, i.e., $BP:PC = 2:3$ and $AQ:QC = 1:4$.

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