Let \(t \in [0, 1]\) be a parameter. Let \(\ell_t\) be the line passing through the points \((t,0)\) and \((0,1-t)\).

For each value of \(a\) we can define the point \(P_t\) where \(\ell_t\) intersects the neighboring lines \(\ell_{t+dt}\).

The collection of points \(P_t\) traces out a curve, as suggested in the drawing above. This curve satisfies the equation \(x^n + y^n = 1\). Determine the value of \(n\).

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