Find the largest positive integer \(n\) such that for all real numbers \(a_1\), \(a_2\), \(\dots , a_{n+1} \), the equation \( a_{n+1} x^2 - 2x \sqrt{ a_1^2 + a_2^2 + \cdots + a_{n+1}^2} + (a_1 + a_2 + \cdots + a_n) = 0\) has real roots.

**Details and assumptions**

Clarification: \( a_1, a_2, \ldots, a_{n+1}\) is **any** set of real numbers. There is no restrictions stated in the problem. They do not need to be integers, nor always positive, nor an arithmetic progression.

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