Find the exact value of the real number $y$ if it exists. Do not use a calculator.
$y=\operatorname{arcsec}\left(-\frac{2 \sqrt{3}}{3}\right)$
Select the correct answer below and, if necessary, fill in the answer box to complete your choice.
A.
$y=\operatorname{arcsec}\left(-\frac{2 \sqrt{3}}{3}\right)=$
(Simplify your answer. Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.)
B.
\[
\operatorname{arcsec}\left(-\frac{2 \sqrt{3}}{3}\right) \text { does not exist }
\]

Step 1 ：The secant function is the reciprocal of the cosine function. The range of the secant function is \((-\infty, -1] \cup [1, \infty)\). The given value \(-\frac{2 \sqrt{3}}{3}\) falls within this range, so the arcsecant of this value does exist.

Step 2 ：To find the exact value of \(y\), we need to find the angle whose secant is \(-\frac{2 \sqrt{3}}{3}\).

Step 3 ：We know that \(\sec(\theta) = \frac{1}{\cos(\theta)}\). So, we need to find the angle \(\theta\) such that \(\cos(\theta) = -\frac{3}{2 \sqrt{3}} = -\frac{\sqrt{3}}{2}\).

Step 4 ：The cosine of \(\frac{\pi}{6}\) is \(\frac{\sqrt{3}}{2}\), but we need the negative of this value. The cosine function is negative in the second and third quadrants. So, the possible values of \(\theta\) are \(\frac{5\pi}{6}\) and \(\frac{7\pi}{6}\).

Step 5 ：However, the range of the arcsecant function is \([0, \frac{\pi}{2}] \cup (\frac{\pi}{2}, \pi] \cup (\pi, \frac{3\pi}{2}] \cup (\frac{3\pi}{2}, 2\pi]\). So, the only possible value of \(y\) is \(\frac{5\pi}{6}\).

Step 6 ：Final Answer: The exact value of the real number \(y\) is \(\boxed{\frac{5\pi}{6}}\).