Show that problem is PSPACE-complete - path in directed graph I have a following problem:
Given $n$ and graph of size $2^n$, and circuit with $2n$ input gates. Directed edge between $k$ and $l$ exists iff only and only we encode $k$ and $l$ as bits and launch circuit on this input and get result $0$.  

Problem: 
Is there exists path from vertex $k$ to vertex $l$ ?  

I can reduce arbitrary problem from $PSPACE$ to above problem. The only thing that I can't do is showing that problem given in exercies belongs to PSPACE.  

Tell me please, if my reasoning is correct (it seems to be too simple):
We know that $PSPACE = NPSPACE$. So we can give non-determinitic algorithm. Algorithm can guess sequentially verticles and check (using circuit) if there exists path).
I can say more, let algorithm guess ansewr (0 or 1).  

Can you help me ?
Given $k$ and $l$ you can check if there is a path between these nodes using DFS. DFS requires polynomial space and time. So,  that problem is in PSPACE.

Update: I'd be glad to take into account the downvoter's opinion and improve the answer.