Stair Game - CSES

I was trying to solve the question Stair Game on cses, and I came across a blog on codeforces, the blogs say that the even position in the stairs does not matter.
When I applied the above idea to the question it got accepted.

using namespace std;
int main(){
    int t; cin>>t;
        int n; cin>>n;
        int xor_ = 0;
        int x; cin>>x;
        for(int i=1;i<n;i++){
            int x; cin>>x;
            //the balls, where k is odd do not matter why?
            xor_ = i&1? xor_^x: xor_;
        cout<<(xor_ ? "first\n": "second\n");

Can someone please give the reasoning behind the idea, why the even stair-cases does not matter?

I think the explanation given in that blog is fine. You can prove it using induction.