Three Tokens and a Bluffing Rate

Ada and Bo play a stripped-down poker game. Three tokens marked $3$, $2$, and $1$ are shuffled; one is dealt face down to each player and the third is set aside unseen. Each player antes $\$1$, so the pot is $\$2$.

Only Ada acts. She may **check**, in which case the tokens are revealed and the higher one takes the pot; or she may **bet $\$1$**, after which Bo either **folds** (Ada takes the $\$2$ pot) or **calls $\$1$**, after which the tokens are revealed and the higher one takes the $\$4$ pot.

Both players know the rules and each other's strategies; neither can be exploited. Find the equilibrium: how often should Ada bet with the $1$, and how often should Bo call with the $2$?