
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
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, where each term is a non-zero real number. Given the following two propositions:
Proposition $p$: $\{a_{n}\}$ is an arithmetic sequence.
Proposition $q$: The equation $\frac{1}{a_{1}a_{2}} + \frac{1}{a_{2}a_{3}} + \dots + \frac{1}{a_{n}a_{n+1}} = \frac{kn+b}{a_{1}a_{n+1}}$ holds for any $n(n \in \mathbb{N}^*)$, where $k$, $b$ are constants.
(1) If $p$ is a sufficient condition for $q$, find the values of $k$ and $b$.
(2) For the $k$ and $b$ in (1), is $p$ a necessary condition for $q$? Explain your reason.
(3) If $p$ is a true proposition, for a given positive integer $n(n > 1)$ and positive number $M$, the sequence $\{a_{n}\}$ satisfies the condition $a_{1}^{2} + a_{n+1}^{2} \leqslant M$, find the maximum value of $S_{n}$.

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