
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 
\[ g_{n}(x) = g_{n-1}\left(\sqrt{n^2 + 2n - x}\right). \]
Let \( M \) be the largest value of \( n \) for which the domain of \( g_n \) is nonempty. For this value of \( M \), the domain of \( g_M \) consists of a single point \(\{d\}\). Compute \( d \).

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