
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 $g_{1}(x) = \sqrt{x^2 + 3}$, and for integers $n \geq 2$, define \[g_{n}(x) = g_{n-1}(\sqrt{n^2 + x}).\] Let $M$ be the largest value of $n$ for which the domain of $g_n$ is nonempty. Determine the point $c$ for which the domain of $g_M$ consists solely of $\{c\}$.

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