
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 $\zeta$ be a complex number such that $\zeta^9 = 1$ and $\zeta \ne 1.$ Let $\gamma = \zeta + \zeta^2 + \zeta^4$ and $\delta = \zeta^3 + \zeta^5 + \zeta^6 + \zeta^7.$ Then $\gamma$ and $\delta$ satisfy the quadratic
\[x^2 + cx + d = 0\]for some real numbers $c$ and $d.$ Find the ordered pair $(c,d).$

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