A strange sequence of polynomials over $\mathbb{F}_{7}$

Question

Let $P=X^{10} +5X^{5}+1$ and $Q=5X^{8}+6X^{3}$ in $\mathbb{F}{7}[X]$. How we can prove this strange relation of Euclid division $$U{2i}P-X^{4}U_{2i-1}^{7}=4.3^{2i-1}Q$$ $$U_{2i+1}P-X^{-4}U_{2i}^{7}=4.3^{2i}Q/X$$ with $U_{1}=X^{2}$?Note that $U_{2i}=[X^{4}U_{2i-1}^{7}/P]$ and $U_{2i+1}=[X^{-4}U_{2i}^{7}/P]$.