Let the smallest value of \(k\in N\) for which the coefficient of \({x}^{3}\) in \((1+x)^{3}+(1+x)^{4}+(1+x)^5+\ldots \l…

Let the smallest value of \(k\in N\) for which the coefficient of \({x}^{3}\) in \((1+x)^{3}+(1+x)^{4}+(1+x)^5+\ldots \ldots+(1+x)^{99}\)\(+(1+k x)^{100}, x \neq 0,\) is \(\left(43n+\frac{101}{4}\right)({}^{100}C_{3})\) for some \(n\in N\) be \(p\). Then the value of \(p+n\) is

[JEE Main 2026, 4 Apr (Shift 1)]