
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 that the complex number $z = m + 2i$ is a root of the equation $x^{2} + 6x + 13 = 0$ $(i$ is the imaginary unit, $m \in \mathbb{R})$.  
$(1)$ Find $|z|$;  
$(2)$ Let $z_{1} = \frac{a - i^{2023}}{\overline{z}}$ (where $\overline{z}$ is the conjugate of $z$), and the point corresponding to the complex number $z_{1}$ is in the third quadrant. Find the range of real number $a$.

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