
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 \( x = \sqrt{\frac{\sqrt{65}}{2} + \frac{5}{2}} \). There exist unique positive integers \( a \), \( b \), \( c \) such that
\[ x^{100} = 3x^{98} + 18x^{96} + 13x^{94} - x^{50} + ax^{46} + bx^{44} + cx^{40}. \]
Find \( a + b + c \).

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