
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
We say that a real  $a\geq-1$  is philosophical if there exists a sequence  $\epsilon_1,\epsilon_2,\dots$ , with  $\epsilon_i \in\{-1,1\}$  for all  $i\geq1$ , such that the sequence  $a_1,a_2,a_3,\dots$ , with  $a_1=a$ , satisfies
      $$ a_{n+1}=\epsilon_{n}\sqrt{a_{n}+1},\forall n\geq1 $$      and is periodic. Find all philosophical numbers.

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