  Write $W^{(n)}$ to mean the $n$-letter word obtained by repeating a fixed
word $W$ and let $R_n$ denote a uniformly random $n$-letter word sampled from
the same alphabet as $W$. We are interested in the average length of the
longest common subsequence between $W^{(n)}$ and $R_n$, which is known to be
$\gamma(W)\cdot n+o(n)$ for some constant $\gamma(W)$. Bukh and Cox recently
developed an interacting particle system, dubbed the frog dynamics, which can
be used to compute the constant $\gamma(W)$ for any fixed word $W$. They
successfully analyzed the simplest case of the frog dynamics to find an
explicit formula for the constants $\gamma(12\cdots k)$. We continue this study
by using the frog dynamics to find an explicit formula for the constants
$\gamma(12\cdots kk\cdots 21)$. The frog dynamics in this case is a variation
of the PushTASEP on the ring where some clocks are identical. Interestingly,
exclusion processes with correlated clocks of this type appear to have not been
analyzed before. Our analysis leads to a seemingly new combinatorial object
which could be of independent interest: frogs with hats!
