
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 $f(z) = (a + bi) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $a = 2, b = 3$, and $\alpha, \gamma$ are complex numbers. Provided that $f(1)$ and $f(i)$ are both real, determine the minimum possible value of $|\alpha| + |\gamma|$.

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