
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 $z = \sqrt{\frac{\sqrt{29}}{2} + \frac{7}{2}}.$  There exist unique positive integers $d,$ $e,$ $f$ such that:
\[
z^{100} = 2z^{98} + 14z^{96} + 11z^{94} - z^{50} + dz^{46} + ez^{44} + fz^{40}.
\]
Find $d + e + f.$

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