
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
Given the sequence $\left\{a_{n}\right\}$ defined by $a_{0} = a_{1} = 1$ and $a_{n} = \sqrt{\frac{n}{2} + a_{n-1} a_{n-2}} \text{ for } n \geq 2$, find all real numbers $t$ such that for any positive integer $n$, $tn < a_{n} < tn + 1$.

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