
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 $\omega$ be a root of the equation $z^4 + 1 = 0$, and specifically consider $\omega = i$ for simplicity. Find the number of ordered pairs $(a, b)$ of integers such that $|a\omega + b| = \sqrt{2}$.

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