
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 $f_{1}(x)=\sqrt{2-x}$, and for integers $n \geq 2$, let \[f_{n}(x)=f_{n-1}\left(\sqrt{n^2 + 1 - x}\right).\] Let $N$ be the largest value of $n$ for which the domain of $f_n$ is nonempty. For this value of $N,$ the domain of $f_N$ consists of a single point $\{c\}.$ Compute $c.$

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