OME 2016 Problem 1

Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$ , $a_2=g_2$ and $a_{10}=g_3$ . Find with proof that, for every positive integer $p$ , there is a positive integer $m$ , such that $g_p=a_m$ .