Two docks are located on an east-west line $2580 \mathrm{ft}$ apart. From dock $A$, the bearing of a coral reef is $65^{\circ} 23^{\prime}$. From dock $B$, the bearing of the coral reef is $335^{\circ} 23^{\prime}$. Find the distance from dock $A$ to the coral reef. The distance from dock $A$ to the coral reef $\mathrm{ft}$. (Round to the nearest integer as needed.)

Step 1 ：We are given two docks located on an east-west line 2580 ft apart. From dock A, the bearing of a coral reef is \(65^{\circ} 23^{\prime}\). From dock B, the bearing of the coral reef is \(335^{\circ} 23^{\prime}\). We are asked to find the distance from dock A to the coral reef.

Step 2 ：This is a problem of trigonometry. We can solve it by using the law of sines. The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

Step 3 ：In this case, we have a triangle formed by the two docks and the coral reef. We know the distance between the two docks and the bearings of the coral reef from each dock, which gives us the angles of the triangle.

Step 4 ：We can use the law of sines to find the distance from dock A to the coral reef. However, the bearings are given as angles from the north, measured clockwise. In the calculation of the angles of the triangle, they are treated as angles from the east, measured counterclockwise. This needs to be corrected.

Step 5 ：After correcting the angles, we find that the distance from dock A to the coral reef is approximately 1645.8185885145401 feet.

Step 6 ：Rounding to the nearest integer, we get the final answer. The distance from dock A to the coral reef is approximately \(\boxed{1646}\) feet.