
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
Let $x = \sqrt{\frac{\sqrt{77}}{3} + \frac{5}{3}}.$  There exist unique positive integers $a,$ $b,$ $c$ such that
\[x^{100} = 3x^{98} + 17x^{96} + 13x^{94} - 2x^{50} + ax^{46} + bx^{44} + cx^{40}.\] Find $a + b + c.$

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