
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
A complex number $z$ satisfies $|z|=\sqrt{2}$, and $z^{2}$ is purely imaginary. If the point corresponding to the complex number $z$ in the complex plane is in the first quadrant, find:

$(1)$ The complex number $z$;

$(2)$ The vectors corresponding to the complex numbers $z$, $\overline{z}$, and $z^{2}$ are $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$, respectively. Given that $(\lambda \overrightarrow{a}+\overrightarrow{b})\perp (\lambda \overrightarrow{b}+\overrightarrow{c})$, find the value of $\lambda$.

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