Step-by-Step Solution

1. Analyze the Assertion:

   Given: $|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}=256$ and $|\vec{b}|=8$

   We know that $|\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}| \sin \theta$ and $\vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta$

2. Substitute the formulas:

   $(|\vec{a}||\vec{b}| \sin \theta)^{2}+(|\vec{a}||\vec{b}| \cos \theta)^{2}=256$

   $|\vec{a}|^{2}|\vec{b}|^{2} \sin ^{2} \theta+|\vec{a}|^{2}|\vec{b}|^{2} \cos ^{2} \theta=256$

   $|\vec{a}|^{2}|\vec{b}|^{2}(\sin ^{2} \theta+\cos ^{2} \theta)=256$

3. Use the trigonometric identity:

   Since $\sin ^{2} \theta+\cos ^{2} \theta=1$, we have $|\vec{a}|^{2}|\vec{b}|^{2}=256$

4. Substitute the value of $|\vec{b}|$:

   $|\vec{a}|^{2}(8)^{2}=256$

   $|\vec{a}|^{2}(64)=256$

5. Solve for $|\vec{a}|$:

   $|\vec{a}|^{2}=\frac{256}{64}=4$

   $|\vec{a}|=\sqrt{4}=2$

6. Conclusion for Assertion:

   The assertion $|\vec{a}|=2$ is correct.

7. Analyze the Reason:

   The reason provides the correct formulas and trigonometric identity used in solving the assertion.

8. Final Answer:

   Both Assertion and Reason are correct, and the Reason is a correct explanation of the Assertion.