
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
Let $p$ be an odd prime and $a$ be any number in the consecutive sequence $2, 3, 4, \cdots, p-3, p-2$. In the sequence $a, 2a, 3a, \cdots, (p-3)a, (p-2)a, (p-1)a$, there must be exactly one number that is congruent to 1 modulo $p$. Let this number be $ia$. Then $i$ is a number in the original sequence and is different from $a$.

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