
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:
\[
\left(1+x+x^{2}+x^{3}\right)^{400} = \sum_{k=0}^{300} c_{4k} x^{4k} + \sum_{k=0}^{299} c_{4k+1} x^{4k+1} + \sum_{k=0}^{299} c_{4k+2} x^{4k+2} + \sum_{k=0}^{299} c_{4k+3} x^{4k+3}
\]
Determine \(\sum_{k=0}^{300} c_{4k}\).

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