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To solve the problem of finding the ratio \( \frac{AI}{AS} \), you need to go through a few steps involving vector analysis and geometry. Here's a breakdown of the main steps:

1. **Understand the Setup:**
- \( M, N, P \) are points on \( AB, AC, AD \) such that \( AM = 2MB, AN = NC, AP = 3PD \).
- \( Q \) is the midpoint of \( BC \).
- \( I \) is the midpoint of \( DQ \).
- \( S \) is the intersection of line \( AI \) with plane \( (MNP) \).

2. **Calculate Vector Positions:**
- Use vector division:
- \( \vec{M} = \frac{2\vec{B} + \vec{A}}{3} \)
- \( \vec{N} = \frac{\vec{C} + \vec{A}}{2} \)
- \( \vec{P} = \frac{3\vec{D} + \vec{A}}{4} \)
- Midpoint of \( BC \):
- \( \vec{Q} = \frac{\vec{B} + \vec{C}}{2} \)
- Midpoint of \( DQ \):
- \( \vec{I} = \frac{\vec{D} + \vec{Q}}{2} = \frac{\vec{D} + \vec{B} + \vec{C}}{4} \)

3. **Finding \( S \):**
- \( S \) lies on line \( AI \) and in plane \( (MNP) \).
- Set up parameters for intersection.

4. **Calculate the Ratio:**
- Use intersection formulas and solve for the parameter that gives the position of \( S \).
- Calculate the ratio \( \frac{AI}{AS} \) using geometric relations or vector distances.

If you're checking the steps from an image, ensure all computations align with the geometry principles and vector math used. The concepts written above should help you verify the logic without repeating parts covered in the problem statement.