
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 $z_{1}=a+i$ and $z_{2}=1-i$, where $a$ is a real number.  
$(1)$ If ${z}_{1}^{2}=-2i$, find the value of the real number $a$.  
$(2)$ If $\frac{{z}_{1}}{{z}_{2}}$ is purely imaginary, find the value of $\frac{{z}_{1}}{{z}_{2}}+{(\frac{{z}_{1}}{{z}_{2}})}^{2}+{(\frac{{z}_{1}}{{z}_{2}})}^{3}+⋯+{(\frac{{z}_{1}}{{z}_{2}})}^{2023}$.

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