
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 in the complex plane, the complex numbers $z_{1}=1+bi\left(b\in R\right)$ and $z_{2}=2-3i$ correspond to points $Z_{1}$ and $Z_{2}$, respectively. The vector $\overrightarrow{{Z_1}{Z_2}}$ is parallel to the real axis.  
$(1)$ Find the value of $b$;  
$(2)$ If the complex number $z={({m+{z_1}})^2}$ corresponds to a point in the third quadrant of the complex plane, find the range of real numbers for $m$.

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