Linear isometry between $\mathcal l_p$ and $\mathcal l_q$ norms in $\mathbb R^n$ implies $p$ and $q$ are conjugate exponents

Question

$\mathbf {The \ Problem \ is}:$ Consider $\mathbb R^n$ with the norms $N_p(x)=\|x\|_p = [\sum_{j=1}^n |x_j|^p]^{1/p} .$

Show that if there is a linear isometry $\phi : (\mathbb R^n,N_p) \to (\mathbb R^n,N_q)$ and $q \neq p,$ then $(1/p + 1/q) =1.$

$\mathbf {My \ approach}:$ I am trying to use the linearity of $\phi$ and if $\{e_i | I=1(|)n\}$ is the standard basis of $\mathbb R^n$, then we can see some values of $\|\phi(e_i)\|_q =1$ for all $i$ ;

But ,I am getting stuck at proving the relation between $p$ and $q$ .

A small hint is very much required at this moment, thanks in advance .