
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 $z=a+i$ (where $a \in \mathbb{R}$), and $(1+2i)z$ is a pure imaginary number.  
(i) Find the complex number $z$.  
(ii) If $\omega = \frac{z}{2-i}$, find the modulus $|\omega|$ of the complex number $\omega$.

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