
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 \( y = \sqrt{\frac{\sqrt{65}}{3} + \frac{5}{3}}.\) There exist unique positive integers \(a,\) \(b,\) \(c\) such that
\[
y^{120} = 3y^{117} + 17y^{114} + 13y^{112} - y^{60} + ay^{55} + by^{53} + cy^{50}.
\]
Find \( a + b + c.\)

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