The correct next step is:

"Using the compass measure between points S and N, draw an arc to the right of line m, centered at T, intersecting the edge of circle P."

Explanation:

To create a line through $P$ that is parallel to line $l$, the process involves constructing corresponding angles. Here's the reasoning:

1. Why use the measure between $S$ and $N$? The distance between $S$ and $N$ corresponds to an angle formed by the intersection of circle $Q$ and lines $l$ and $m$. This angle is essential to replicate at point $P$.

2. Why draw the arc centered at $T$? The arc centered at $T$, with the same radius as the arc between $S$ and $N$, ensures that the angles formed at $T$ replicate the geometry of the angles at $Q$. This is critical for constructing a parallel line.

3. What happens when the arc intersects circle $P$? The point of intersection of this arc with circle $P$ will be the desired point $R$. Connecting $P$ and $R$ creates a line parallel to $l$.

Would you like further clarification or visual examples of these steps?

---

Related Questions:

1. Why does the construction method ensure that the new line is parallel to the original line $l$?
2. Can you describe the role of corresponding angles in this construction?
3. Why is it necessary to use both circles in this construction?
4. What is the significance of using point $T$ for this arc instead of any other point?
5. How would the construction differ if the original line $l$ were curved instead of straight?

Tip:

When constructing parallel lines using geometric tools, always verify the accuracy of your angle replication to avoid subtle misalignments.