A game is played on the board shown. In this game, a player can move three places in any direction (up, down, right or left) and then can move two places in a direction perpendicular to the first move. If a player starts at $S$, which position on the board ($P, Q, R, T$, or $W$) cannot be reached through any sequence of moves?

\[ \begin{tabular}{|c|c|c|c|c|}\hline  &  & P & & \\ \hline  & Q & & R &\\ \hline  &  & T & & \\ \hline S & & & & W\\ \hline\end{tabular} \]

$\textbf{(A)}\ P \qquad \textbf{(B)}\ Q \qquad \textbf{(C)}\ R \qquad \textbf{(D)}\ T \qquad \textbf{(E)}\ W$
To determine which position on the board cannot be reached from $S$ using the given movement rules, we need to analyze the possible moves step-by-step.

1. **Initial Position and Moves**:
   - The player starts at position $S$.
   - The player can move three places in any direction (up, down, right, or left).
   - After the first move, the player can move two places in a direction perpendicular to the first move.

2. **Possible Moves from $S$**:
   - From $S$, moving three places up is not possible as it would go off the board.
   - Moving three places down is also not possible as it would go off the board.
   - Moving three places left is not possible as it would go off the board.
   - Moving three places right lands the player at position $W$.

3. **Analyzing Moves from $W$**:
   - From $W$, moving two places up lands the player at position $T$.
   - Moving two places down is not possible as it would go off the board.
   - Moving two places left lands the player at position $S$.
   - Moving two places right is not possible as it would go off the board.

4. **Other Possible Moves from $S$**:
   - Moving three places up is not possible.
   - Moving three places down is not possible.
   - Moving three places left is not possible.
   - Moving three places right lands the player at $W$.

5. **Checking Other Positions**:
   - To reach $P$, the player would need to move three places up and then two places left, which is not possible.
   - To reach $Q$, the player would need to move three places up and then two places right, which is not possible.
   - To reach $R$, the player would need to move three places up and then two places right, which is not possible.
   - To reach $T$, the player would need to move three places right and then two places up, which is possible.

6. **Conclusion**:
   - The only position that cannot be reached from $S$ through any sequence of moves is $P$.

The final answer is $\boxed{P}$.