Spectral norm of Jordan basis

Question

Suppose that $A \in \mathbb{R}^{n \times n}$ with $\|A\|_2 \leq R$. and let $A = P J P^{-1}$ be a Jordan canonical form. Are there any upper bounds on the norms $\|P\|_2$ and $\|P^{-1}\|_2$ in terms of $R$ or other quantities related to $A$? If $A$ is symmetric with distinct eigenvalues, then $P$ is the eigenvector matrix and $\|P\|_2, \|P^{-1}\|_2 = 1$. This is related to the question I asked earlier.