
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
If the first term of an arithmetic sequence is $C_{5n}^{11-2n}-P_{11-3n}^{2n-2}$, and the common difference is the constant term in the expansion of $\left( \frac{5}{2x}-\frac{2}{5}\sqrt[3]{{x}^{2}} \right)^{m}$, where $m$ is the remainder of ${{77}^{77}}-15$ divided by $19$, then how many terms of this sequence have the maximum sum? And what is this maximum value?

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