
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
A sequence  $a_n$  of real numbers satisfies  $a_1=1$ ,  $a_2=0$ , and  $a_n=(S_{n-1}+1)S_{n-2}$  for all integers  $n\geq3$ , where  $S_k=a_1+a_2+\dots+a_k$  for positive integers  $k$ . What is the smallest integer  $m>2$  such that  $127$  divides  $a_m$ ?

*2020 CCA Math Bonanza Individual Round #9*

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