
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 \(\omega\) be a complex number such that \(\omega^5 = 1\) and \(\omega \ne 1.\) Define \(\alpha = \omega + \omega^2\) and \(\beta = \omega^3 + \omega^4.\) Then, \(\alpha\) and \(\beta\) satisfy the quadratic
\[x^2 + ax + b = 0\]
for some real numbers \(a\) and \(b\). Determine the ordered pair \((a, b)\).

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