
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
In the process of summing a series, the method of splitting and canceling terms is commonly used. For example, when calculating $S_{n}=1+2+3+\cdots +n$, we can choose to manipulate the general term as follows: $a_{n}=n=-\frac{1}{2}\left[\left(n-1\right)n-n\left(n+1\right)\right]$. Thus, we can find $S_{n}=-\frac{1}{2}\left[0\times 1-1\times 2+1\times 2-2\times 3+\cdots +\left(n-1\right)n-n\left(n+1\right)\right]=\frac{n(n+1)}{2}$. By analogy, calculate $S_{n}=1\times 2+2\times 3+\cdots +n\left(n+1\right)=$______, and derive the formula for the sum of squares of natural numbers $1^{2}+2^{2}+3^{2}+\cdots +n^{2}=\_\_\_\_\_\_$.

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