
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
Find all  $C\in \mathbb{R}$  such that every sequence of integers  $\{a_n\}_{n=1}^{\infty}$  which is bounded from below and for all  $n\geq 2$  satisfy   $$ 0\leq a_{n-1}+Ca_n+a_{n+1}<1 $$   is periodic.

Proposed by Navid Safaei

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