
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 a function \( f \) be defined by \( f(z) = (5 + 2i) z^2 + \alpha z + \gamma \) for all complex numbers \( z \), where \( \alpha \) and \( \gamma \) are complex numbers, and \( i^2 = -1 \). Assume \( f(1) \) and \( f(-i) \) are both real numbers. Determine the smallest possible value of \( | \alpha | + |\gamma | \).

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