
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 $z_{1}=1+ai$ (where $a\in R$ and $a \lt 0$, and $i$ is the imaginary unit), and $z_1^2$ is purely imaginary.  
$(1)$ Find the value of the real number $a$;  
$(2)$ If ${z_2}=\frac{{{z_1}}}{{1+i}}+2$, find the modulus $|z_{2}|$ of the complex number $z_{2}$.

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