
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 \( f(z) = z^2 - 2iz + 2 \). How many complex numbers \(z \) are there such that \( \text{Im}(z) > 0 \) and both the real and the imaginary parts of \(f(z)\) are integers with absolute value at most \(10\)?

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