
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 $g_{1}(x) = \sqrt{2-x}$, and for integers $n \geq 2$, define 
\[ g_{n}(x) = g_{n-1}(\sqrt{n^3 - x}). \]
Determine the maximum integer $N$ for which the domain of $g_n$ is non-empty. For this value of $N$, find the domain consisting of a single point $\{d\}$, and compute $d$.

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