
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
Let $m$ be a real number, and let the complex numbers be $z_{1}=1+2\mathrm{i}$ and $z_{2}=m+3\mathrm{i}$ (where $\mathrm{i}$ is the imaginary unit). If $z_{1} \overline{z_{2}}$ is a purely imaginary number, then the value of $\left|z_{1}+z_{2}\right|$ is ____ .

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