
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
(12 points) By calculation, one can obtain the following equations:  
$2^2-1^2=2\times 1+1$  
$3^2-2^2=2\times 2+1$  
$4^2-3^2=2\times 3+1$  
...  
$(n+1)^2-n^2=2\times n+1$  
By summing the above equations, we get: $(n+1)^2-1^2=2\times (1+2+3+...+n)+n$  
Therefore, the sum of the first n natural numbers, $1+2+3+...+n=$  
By analogy with the above method: please calculate the value of $1^2+2^2+3^2+...+n^2$.

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