
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 \( y = \sqrt{\frac{\sqrt{37}}{3} + \frac{5}{3}} \). There exist unique positive integers \( d, e, f \) such that
\[ y^{50} = 3y^{48} + 10y^{45} + 9y^{43} - y^{25} + dy^{21} + ey^{19} + fy^{15}. \]
Find \( d + e + f \).

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