
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 \( n \geq 2 \), let \[ g_n(x) = g_{n-1}(\sqrt{(n+1)^2 - x}). \] 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