
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 $b$, a new number $c$ can be derived using the rule $c=ab+a+b$. Among the three numbers $a$, $b$, and $c$, select the two larger numbers and apply the same rule to generate a new number. This process is considered one operation.
1. After two operations on the positive numbers 1 and 2, the resulting number is \_\_\_\_\_\_.
2. If $p > q > 0$, and the number obtained after five operations is $(q+1)^{m}(p+1)^{n}-1$ ($m$, $n$ are positive integers), then $m+n=$ \_\_\_\_\_\_.

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