
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 $f : \mathbb{C} \to \mathbb{C}$ be defined by $f(z) = z^2 + iz + 2$. Determine the number of complex numbers $z$ such that $\text{Im}(z) > 0$ and both the real and the imaginary parts of $f(z)$, say $a+bi$, are integers with $|a|, |b| \leq 5$.

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