
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
Given the function  $ f(z) \equal{} (4 \plus{} i) z^2 \plus{} \alpha z \plus{} \gamma$  for all complex numbers  $ z$, where  $ \alpha$  and  $ \gamma$  are complex numbers with  $ i^2 \equal{} \minus{} 1$, find the smallest possible value of  $ |\alpha| \plus{} |\gamma|$.

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