
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 the largest constant \( M > 0 \) such that for any positive integer \( n \), there exist positive sequences \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \) satisfying:

1. \(\sum_{k=1}^{n} b_{k}=1\),
   
   \(2 b_{k} \geq b_{k-1}+b_{k+1}\) for \( k = 2, 3, \ldots, n-1 \);

2. \(a_k^2 \leq 1 + \sum_{i=1}^{k} a_i b_i\) for \( k = 1, 2, \ldots, n \) with \( a_n = M \).

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