
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 infinite sequences of positive real numbers \( X_1, X_2, \cdots \) such that for \( n \in \mathbf{Z}_{+} \),
$$
X_{n+2} = \frac{1}{2}\left(\frac{1}{X_{n+1}} + X_{n}\right).
$$

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