Solve the equation for exact solutions over the interval $[0,2 \pi)$.
\[
3 \cot x+2=5
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is
(Type an exact answer, using $\pi$ as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B. The solution is the empty set

Step 1 ：We can write the equation as \[3 \frac{\cos x}{\sin x} + 2 = 5.\]

Step 2 ：Then \[3 \frac{\cos x}{\sin x} = 3,\] so \[\frac{\cos x}{\sin x} = 1.\]

Step 3 ：This implies \[\cos x = \sin x.\]

Step 4 ：We know that \[\sin^2 x + \cos^2 x = 1,\] so \[2 \sin^2 x = 1,\] which gives \[\sin^2 x = \frac{1}{2}.\]

Step 5 ：This equation factors as \[(\sin x - \frac{\sqrt{2}}{2})(\sin x + \frac{\sqrt{2}}{2}) = 0,\] so \[\sin x = \frac{\sqrt{2}}{2}\] or \[\sin x = -\frac{\sqrt{2}}{2}.\]

Step 6 ：However, if \[\sin x = -\frac{\sqrt{2}}{2},\] then \[\cos x = -\frac{\sqrt{2}}{2},\] which means \[\cot x\] is negative. So \[\sin x = \frac{\sqrt{2}}{2},\] which leads to the \[\boxed{2}\] solutions \[x = \frac{\pi}{4}\] and \[x = \frac{7 \pi}{4}.\] We check that both solutions work.