
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 $y = \sqrt{\frac{\sqrt{75}}{2} + \frac{5}{2}}.$ There exist unique positive integers $d,$ $e,$ $f$ such that
\[y^{100} = 3y^{98} + 18y^{96} + 15y^{94} - y^{50} + dy^{46} + ey^{44} + fy^{40}.\] Find $d + e + f.$

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