
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 \( y = \sqrt{\frac{\sqrt{61}}{2} + \frac{5}{2}} \). There exist unique positive integers \( p, q, r \) such that
\[
y^{100} = 2y^{98} + 16y^{96} + 13y^{94} - y^{50} + py^{46} + qy^{44} + ry^{40}.
\]
Find \( p + q + r \).

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