
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 a complex number $z=2+bi(b \gt 0$, $i$ is the imaginary unit), and $z^{2}$ is a purely imaginary number.  
$(1)$ Find the value of the real number $b$;  
$(2)$ If the complex number $\omega=\frac{z}{1-i}$, find the modulus of $\omega$.

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