
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 two positive numbers $a$ and $b$, they can be extended to a new number $c$ by the rule $c = ab + a + b$. From the three numbers $a$, $b$, and $c$, take the two larger numbers and apply the same rule to extend to a new number. Continue this process, and denote each extension as one operation.

(1) The number obtained after two extensions starting with positive numbers 1 and 2 is _______.

(2) If $p > q > 0$, and after five operations the extended number is $(q+1)^m(p+1)^n - 1$ (where $m$, $n$ are positive integers), then $m + n =$ _______.

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