
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 function $g : \mathbb{Z} \to \mathbb{Z}$ satisfies:
\begin{align*}
g(x+5)-g(x) &= 10x+30, \\
g(x^2-2) &= (g(x)-x)^2+x^2-4
\end{align*}
for all integers $x.$ Determine the ordered pair $(g(0), g(1))$.

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