
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
Identify the complete squares: 
\[
\begin{cases}
(x-2)^{2}+(y+1)^{2}=5, \\
(x-2)^{2}+(z-3)^{2}=13, \\
(y+1)^{2}+(z-3)^{2}=10.
\end{cases}
\]

Add all the equations: 
\[
\begin{cases}
(x-2)^{2}+(y+1)^{2}=5, \\
(x-2)^{2}+(z-3)^{2}=13, \\
(x-2)^{2}+(y+1)^{2}+(z-3)^{2}=14.
\end{cases}
\]

By subtracting the first and second equations from the third, obtain:

\[
\begin{cases}
(x-2)^{2}=4 \\
(z-3)^{2}=9 \\
(y+1)^{2}=1
\end{cases}
\]

Solve the quadratic equations and find all possible solutions. The system has eight solutions.

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