
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 $z$ is an imaginary number, if $\frac{z}{{1+i}}\in R$, and $-1 \lt \frac{z}{1+i} \lt 2$.  
$(1)$ Find the range of the real part of $z$;  
$(2)$ Find the range of $|z+3\overline{z}|$.

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