
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 complex numbers $z_{1}=(2x+1)+2i$, $z_{2}=-x-yi$ $(x,y\in\mathbb{R})$

$(1)$ If $z_{1}+z_{2}=0$, find the value of $x^{2}-y^{2}$;  

$(2)$ If the complex number $(1+i) \cdot z_{1}$ is purely imaginary, find the modulus of the complex number $z_{1}$, $|z_{1}|$.

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