
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 \( p \geq 5 \) be a prime number. For a positive integer \( k \), define \( R(k) \) as the remainder when \( k \) is divided by \( p \) (i.e., \( 0 \leq R(k) \leq p-1 \)). Find all positive integers \( a \) such that for every \( m \in \{1, 2, \ldots, p-1\} \), the inequality \( m + R(ma) > a \) holds.

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