
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 $\{a_{n}\}$ with the first term $a_{1}=2$, and the sum of its first $n$ terms denoted as $S_{n}$, satisfying $S_{n}+S_{n-1}=\left(n+1\right)\left(n+2\right)$ for $n\geqslant 2$.<br/>$(1)$ Find $S_{21}$;<br/>$(2)$ Let ${c_n}=\frac{{{{({-1})}^n}n}}{{{S_n}}}$, determine the maximum term of the sequence $\{c_{n}\}$.

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