
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 that the population is divided into several layers, and through stratified random sampling, the sample data extracted from one layer is $x_{1}$, $x_{2}$, $\ldots $, $x_{n}$, with mean and variance being $\overline{x}$, $s^{2}$, respectively. Let the overall sample mean be $\overline{\omega}$, find the value of $\sum_{i=1}^n{{{({{x_i}-\overline{\omega}})}^2}}$.

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