= Solution
Every <projective line> on the <smooth algebraic surface> through $P$ lies in the <projective tangent plane> $T_PS$, since its tangent direction is contained in the surface's <Zariski tangent space>. The <plane section> $S\cap T_PS$ is a nonzero <plane cubic>. It cannot be the entire plane, since a <smooth cubic surface> is irreducible and has no plane component.
Each distinct <projective line> through $P$ is therefore a distinct degree-one factor of the cubic restricted to $T_PS$. A degree-three <polynomial> has at most three such factors. Hence there are \b[at most three lines through any point]. This is the <lines through a point of a smooth cubic surface> bound; equality is allowed when the tangent section consists of three concurrent lines.
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