Function $p$ is defined by $p(x)=-c^x+d$, where $c$ and $d$ are constants. In the $xy$-plane, the graph of $y=p(x)+1$ has a $y$-intercept at $(0,\frac{11}{6})$. The product of $c$ and $d$ is $\frac{5}{2}$. What is the value of $c$?