
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
Find all natural numbers \( n \) such that the equation \( a_{n+1} x^{2} - 2 x \sqrt{a_{1}^{2} + a_{2}^{2} + \cdots + a_{n+1}^{2}} + a_{1} + a_{2} + \cdots + a_{n} = 0 \) has a real solution for all real numbers \( a_{1}, a_{2}, \cdots, a_{n+1} \).

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