Let $ABCD$ be a square and {$X_t, t \in \mathbb{N}$} a discrete random walk on this square (i.e the state space is {$A,B,C,D$}). Let
$$\tau_D := \min\{ t\geq0 : X_t = D \}$$ and $$\tau_C := \min\{ t\geq0 : X_t = C \}$$
How to calculate:
$$P_x(\tau_C \leq s \cap \tau_C > \tau_D )$$, where $s \in \mathbb{N}$ and $x \in$ {$A,B,C,D$} and $P_x(.) = P(.|X_0 = x)$
My attempt was: $$P_x(\tau_C \leq s \cap \tau_C > \tau_D ) = P_x(\tau_C \leq s | \tau_C > \tau_D )P_x(\tau_C > \tau_D)$$
But I can't find a way to continue after it.