
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
 $(x_{n})_{-\infty<n<\infty}$  is a sequence of real numbers which satisfies  $x_{n+1}=\frac{x_{n}^2+10}{7}$  for every  $n \in \mathbb{Z}$ . If there exist a real upperbound for this sequence, find all the values  $x_{0}$  can take.

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