Shrinking a Point Into Every Unit Ball

The point $\mathbf{p} = (0.8,\ 0.6)$ lies outside at least one of the three unit balls in $\mathbb{R}^2$ — the L1 diamond, the L2 circle, and the L∞ square, each of radius $1$.

You shrink it toward the origin by scaling:

$\mathbf{p}' = s\,\mathbf{p}, \qquad s > 0$

What is the largest value of $s$ for which $\mathbf{p}'$ lies inside or on all three unit balls at once?