
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
The real number sequence \(\{a_n\}\) satisfies
\[1=a_0 \leqslant a_1 \leqslant a_2 \leqslant \cdots,\]

The sequence \(\{b_n\}\) satisfies
\[b_n = \sum_{k=1}^{n}\left(1-\frac{a_{k-1}}{a_k}\right) \frac{1}{\sqrt{a_k}}, \quad (n=1,2, \cdots).\]

Find the largest real number \(C\) such that for any real number \(x \in [0, C)\), there exists a sequence \(\{a_n\}\) satisfying the condition (1), such that the corresponding sequence \(\{b_n\}\) has infinitely many terms greater than \(x\).

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