
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
A function $f$ is defined by $f(z) = (3 + 2i)z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma are complex numbers and $i^2 = -1$. It is given that the outputs of $f(1)$ and $f(i)$ are both real. Determine the smallest possible value of $|\alpha| + |\gamma|$.

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