
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  $n\ge 2$ . 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}
user