1. Identify the given elements and their relationships:
- Triangle $ABC$ with sides $AB = 6$, $BC = 5$, and $AC = 7$.
- Tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X$.
- Point $Z$ on the circumcircle of $ABC$.
- $Y$ is the foot of the perpendicular from $X$ to $CZ$.
- $K$ is the intersection of the circumcircle of $BCY$ with line $AB$.
- $YZ = 3CY$.

2. Use the Power of a Point theorem:
- The power of a point $A$ with respect to the circumcircle of $BCY$ is given by:
$$\text{Pow}(A, \omega) = \text{Pow}(A, \Omega) + k(\text{Pow}(B, \omega, \Omega) + (1-k)\text{Pow}(T, \omega, \Omega))$$
- Here, $\Omega$ is the circumcircle of $ABC$ and $\omega$ is the circumcircle of $BCY$.

3. **Calculate the power of point $A$ with respect to $\Omega$:**
- Since $A$ lies on the circumcircle $\Omega$, $\text{Pow}(A, \Omega) = 0$.

4. **Determine the power of point $T$ with respect to $\omega$ and $\Omega$:**
- Let $CZ \cap AB = T$.
- By the Power of a Point theorem:
$$\text{Pow}(T, \omega) = TY \cdot TC$$
$$\text{Pow}(T, \Omega) = TZ \cdot TC$$

5. **Use the given length condition $YZ = 3CY$:**
- Let $CY = y$, then $YZ = 3y$.
- Therefore, $CZ = CY + YZ = y + 3y = 4y$.

6. **Calculate the lengths $TY$ and $TZ$:**
- Since $Y$ is the foot of the perpendicular from $X$ to $CZ$, $TY = TZ$.
- Using the given ratio, $TY = \frac{1}{4}CZ = \frac{1}{4}(4y) = y$.

7. Substitute the values into the power of point expressions:
$$\text{Pow}(T, \omega) = TY \cdot TC = y \cdot TC$$
$$\text{Pow}(T, \Omega) = TZ \cdot TC = y \cdot TC$$

8. **Calculate the power of point $A$ with respect to $\omega$:**
$$\text{Pow}(A, \omega) = (1-k)(\text{Pow}(T, \omega) - \text{Pow}(T, \Omega)) = (1-k)(y \cdot TC - y \cdot TC) = 0$$

9. **Determine the value of $k$:**
- Since $k = -\frac{AT}{TB}$, we need to find the lengths $AT$ and $TB$.

10. **Use the Law of Sines to find $ZC$:**
$$ZC = \frac{70}{11}$$
$$ZY = \frac{105}{22}$$

11. **Find $TC$ using the ratio lemma:**
$$AT = \frac{343}{78}$$
$$AZ = \frac{49}{11}$$
$$TC = \frac{385}{78}$$

12. **Substitute the values to find $\text{Pow}(A, \omega)$:**
$$\text{Pow}(A, \omega) = \frac{441}{5}$$

13. **Calculate $AK$:**
$$AK = \boxed{\frac{147}{10}}$$