Ada rolls a standard $4$-sided die $5$ times. The probability that the die lands on at most two distinct sides can be written as $ \frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$
1. **Calculate the probability of landing on exactly one distinct side:**
   - There are 4 possible sides on a 4-sided die.
   - The probability of landing on the same side in all 5 rolls is:
     \[
     4 \cdot \left(\frac{1}{4}\right)^5 = 4 \cdot \frac{1}{1024} = \frac{4}{1024} = \frac{1}{256}
     \]

2. **Calculate the probability of landing on exactly two distinct sides:**
   - Choose 2 sides out of 4, which can be done in \(\binom{4}{2} = 6\) ways.
   - For each pair of sides, we need to distribute 5 rolls among these 2 sides.
   - The number of ways to distribute 5 rolls among 2 sides such that both sides appear at least once is given by:
     \[
     \binom{5}{1} + \binom{5}{2} + \binom{5}{3} + \binom{5}{4} = 5 + 10 + 10 + 5 = 30
     \]
   - The total number of outcomes for 5 rolls is \(4^5 = 1024\).
   - Therefore, the probability of landing on exactly two distinct sides is:
     \[
     6 \cdot \frac{30}{1024} = \frac{180}{1024} = \frac{45}{256}
     \]

3. **Combine the probabilities:**
   - The total probability of landing on at most two distinct sides is the sum of the probabilities of landing on exactly one distinct side and exactly two distinct sides:
     \[
     \frac{1}{256} + \frac{45}{256} = \frac{1 + 45}{256} = \frac{46}{256} = \frac{23}{128}
     \]

4. **Express the probability in the form \(\frac{A}{B}\) where \(A\) and \(B\) are relatively prime:**
   - Here, \(A = 23\) and \(B = 128\).

5. **Calculate \(1000A + B\):**
   \[
   1000 \cdot 23 + 128 = 23000 + 128 = 23128
   \]

The final answer is \(\boxed{23128}\).