
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 the complex number $z=3+bi \ (b\in \mathbb{R})$, and $(1+3i) \cdot z$ is a pure imaginary number.  
(a) Find the complex number $z$ and its conjugate $\overline{z}$.  
(b) If $\omega = \frac{z}{2+i}$, calculate the modulus $|\omega|$ of the complex number $\omega$.

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