
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
Consider the function \( f \) defined by \( f(z) = (3 + 2i) z^2 + \beta z + \delta \) for all complex numbers \( z \), where \( \beta \) and \( \delta \) are complex numbers. Assume \( f(2) \) and \( f(-i) \) are real numbers. Determine the smallest possible value of \( |\beta| + |\delta| \).

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