
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 a sequence \(\left\{a_{n}\right\}\) of nonzero terms with the sum of the first \(k\) terms denoted by \(S_{k}\), where
\[ S_{k} = \frac{1}{2} a_{k} a_{k+1} \quad (k \in \mathbb{N}_{+}), \]
and \(a_{1} = 1\).

(1) Find the general term formula of the sequence \(\left\{a_{n}\right\}\).

(2) For any given positive integer \(n (n \geqslant 2)\), the sequence \(\left\{b_{n}\right\}\) satisfies:
\[ \frac{b_{k+1}}{b_{k}} = \frac{k-n}{a_{k+1}} \quad (k=1,2,3, \cdots, n), \quad b_{1}=1. \]
Find the sum \(b_{1}+b_{2}+\cdots+b_{n}\).

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