
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
The equation $a^{12}xz - a^{10}z - a^9x = a^8(b^6 - 1)$ is equivalent to $(a^mx - a^n)(a^pz - a^3) = a^8b^6$ for some integers $m$, $n$, and $p$. Find the value of $mnp$.

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