
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 complex numbers $z_{1}=1-2i$ and $z_{2}=4+3i$.  
$(1)$ Find $z_{1}z_{2}$;  
$(2)$ If $|z|=2$, and the imaginary part of complex number $z$ is equal to the real part of $3z_{1}-z_{2}$, and the complex number $z$ corresponds to a point in the third quadrant of the complex plane, find the complex number $z$.

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