
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
The ancient mathematician Yang Hui, building upon Shen Kuo's work on piled sphere counts, considered: if spheres of identical size are stacked to form a shape similar to a regular quadrangular frustum, with the top layer composed of \(a \times a\) spheres, Yang Hui provided a formula to calculate the total number of spheres in the stack as follows: \(S = \frac{n}{3}(a^2 + b^2 + ab + \frac{b-a}{2})\). Based on the information provided, we can obtain \(1^2 + 2^2 + \ldots + n^2 =\) ______.

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