Very ample polarized self maps extend to projective space

Let $X$ be a projective variety defined over an infinite field, equipped with a line bundle $L$, giving an embedding of $X$ into $\mb{P}^m$ and let $ϕ: X \to X$ be a morphism such that $ϕ^*L \cong L^{\otimes q}, q\geq 2$. Then there exists an integer $r>0$ extending $ϕ^r$ to $\mb{P}^m$.

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