I'm trying to make a PDA that accepts the language $\{ w001w^\text{R} : w = \{0,1\}^* \}$ by empty stack. (Here $w^\text{R}$ denotes the reverse of the string $w$.) Our stack symbol s $\#$.
I've come up with this so far, but I don't know how to handle the $001$ in the middle of $w$ and $w^\text{r}$. Any suggestions?
Start by pushing $0,1$ onto the stack: $(q_0,0,\#) \rightarrow (q_0,0\#)$ and $(q_0,1,\#) \rightarrow (q_0, 1\#)$
Continue to push $0,1$ onto stack: $(q_0,0,0) \rightarrow (q_0,00)$, $(q_0,0,1) \rightarrow (q_0, 01)$, $(q_0,1,0) \rightarrow (q_0,10)$, $(q_0,1,1) \rightarrow (q0, 11)$
Handle $001$ and transition to $q_1$: I don't know how to do this
Pop $0,1$ off stack: $(q_1,0,0) \rightarrow (q_1,E)$, $(q_1,1,1) \rightarrow (q_1,E)$
If stack is empty after this point, than accept language.