1. We start with the given expression for $a$:
$$a = \frac{10^{2n} - 1}{3(10^n + 1)}$$

2. We simplify the numerator $10^{2n} - 1$ using the difference of squares:
$$10^{2n} - 1 = (10^n)^2 - 1^2 = (10^n - 1)(10^n + 1)$$

3. Substituting this back into the expression for $a$:
$$a = \frac{(10^n - 1)(10^n + 1)}{3(10^n + 1)}$$

4. Since $10^n + 1$ is a common factor in both the numerator and the denominator, we can cancel it out:
$$a = \frac{10^n - 1}{3}$$

5. Next, we analyze the form of $10^n - 1$. This is a number consisting of $n$ digits of 9. For example, if $n = 3$, then $10^3 - 1 = 999$.

6. Dividing $10^n - 1$ by 3, we get a number consisting of $n$ digits of 3. For example, if $n = 3$, then $\frac{999}{3} = 333$.

7. We are given that the sum of the digits of $a$ is 567. Since $a$ consists of $n$ digits of 3, the sum of the digits of $a$ is:
$$3n = 567$$

8. Solving for $n$:
$$n = \frac{567}{3} = 189$$

The final answer is $\boxed{189}$