
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 an odd integer $p > 5$, check whether:
- $(p+1)^{\frac{1}{2}(p-3)} - 1$ is divisible by $p+2$,
- $(p-1)^{\frac{1}{2}(p-1)} - 1$ is divisible by $p-2$.
A) Both statements are correct.
B) Only "$(p+1)^{\frac{1}{2}(p-3)} - 1$ is divisible by $p+2$" is correct.
C) Only "$(p-1)^{\frac{1}{2}(p-1)} - 1$ is divisible by $p-2$" is correct.
D) Neither statement is correct.

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