
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

Simplify the expression

$$
K = x^{n} + x^{n-1} + 2x^{n-2} + 2x^{n-3} + 3x^{n-4} + \ldots + 3x^{4} + 2x^{3} + 2x^{2} + x + 1
$$

in which the coefficients increment by 1 after every pair of equal coefficients from both sides towards the middle.

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