
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
The Fibonacci sequence is defined as follows: \( F_{0}=0, F_{1}=1 \), and \( F_{n}=F_{n-1}+F_{n-2} \) for all integers \( n \geq 2 \). Find the smallest positive integer \( m \) such that \( F_{m} \equiv 0 \pmod{127} \) and \( F_{m+1} \equiv 1 \pmod{127} \).

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