
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

a) Find all positive integers \( g \) with the following property: for each odd prime number \( p \) there exists a positive integer \( n \) such that \( p \) divides the two integers
\[ g^{n} - n \text{ and } g^{n+1} - (n+1) \text{.} \]

b) Find all positive integers \( g \) with the following property: for each odd prime number \( p \) there exists a positive integer \( n \) such that \( p \) divides the two integers
\[ g^{n} - n^{2} \quad \text{ and } \quad g^{n+1} - (n+1)^{2} \text{.} \]

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