
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
Define $g_{1}(x)=\sqrt{1-x}$, and for integers $n \geq 2$, let \[g_{n}(x)=g_{n-1}\left(\sqrt[3]{n^3 - x}\right).\] Determine the largest value of $n$, denoted as $M$, for which the domain of $g_n$ is nonempty. For this value of $M$, if the domain of $g_M$ consists of a single point $\{d\}$, compute $d$.

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