
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
Let $f(x)$ be a function defined on $R$ such that $f(0) = 2$, and for any $x, y \in R$, we have $f(xy+1) = f(x) \cdot f(y) - 2f(y) - 2x + 3$.
$(1)$ Find the analytical expression of the function $f(x)$.
$(2)$ Let $g(x) = x - \sqrt{f(x)}$. Find the range of the function $g(x)$.

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