
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 $z$ satisfies $z=\left(-1+3i\right)\left(1-i\right)-4$.  
$(1)$ Find the conjugate complex number of $z$;  
$(2)$ If $\omega =z+ai$, and the modulus of the complex number $\omega$ corresponding to the vector is not greater than the modulus of the vector corresponding to the complex number $z$, find the range of the real number $a$.

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