Construct a geometric gure in a sequence of steps. In step $1$, begin with a $4\times 4$ square. In step $2$, attach a $1\times 1$ square onto the each side of the original square so that the new squares are on the outside of the original square, have a side along the side of the original square, and the midpoints of the sides of the original square and the attached square coincide. In step $3$, attach a $\frac14\times  \frac14$ square onto the centers of each of the $3$ exposed sides of each of the $4$ squares attached in step $2$. For each positive integer $n$, in step $n + 1$, attach squares whose sides are $\frac14$ as long as the sides of the squares attached in step n placing them at the centers of the $3$ exposed sides of the squares attached in step $n$. The diagram shows the gure after step $4$.
If this is continued for all positive integers $n$, the area covered by all the squares attached in all the steps is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
[img]https://cdn.artofproblemsolving.com/attachments/2/1/d963460373b56906e93c4be73bc6a15e15d0d6.png[/img]
1. **Step 1: Calculate the area of the initial square.**
   - The initial square is a \(4 \times 4\) square.
   - The area of the initial square is:
     \[
     4 \times 4 = 16
     \]

2. **Step 2: Calculate the area of the squares added in step 2.**
   - In step 2, four \(1 \times 1\) squares are added, one on each side of the initial square.
   - The area of each \(1 \times 1\) square is:
     \[
     1 \times 1 = 1
     \]
   - Since there are 4 such squares, the total area added in step 2 is:
     \[
     4 \times 1 = 4
     \]

3. **Step 3: Calculate the area of the squares added in step 3.**
   - In step 3, squares of side length \(\frac{1}{4}\) are added to the centers of the 3 exposed sides of each of the 4 squares added in step 2.
   - The area of each \(\frac{1}{4} \times \frac{1}{4}\) square is:
     \[
     \left(\frac{1}{4}\right)^2 = \frac{1}{16}
     \]
   - There are \(4 \times 3 = 12\) such squares, so the total area added in step 3 is:
     \[
     12 \times \frac{1}{16} = \frac{12}{16} = \frac{3}{4}
     \]

4. **Step 4: Calculate the area of the squares added in step 4.**
   - In step 4, squares of side length \(\frac{1}{16}\) are added to the centers of the 3 exposed sides of each of the 12 squares added in step 3.
   - The area of each \(\frac{1}{16} \times \frac{1}{16}\) square is:
     \[
     \left(\frac{1}{16}\right)^2 = \frac{1}{256}
     \]
   - There are \(12 \times 3 = 36\) such squares, so the total area added in step 4 is:
     \[
     36 \times \frac{1}{256} = \frac{36}{256} = \frac{9}{64}
     \]

5. **Step 5: Generalize the pattern and sum the infinite series.**
   - The side length of the squares added in step \(n\) is \(\left(\frac{1}{4}\right)^{n-1}\).
   - The area of each square added in step \(n\) is:
     \[
     \left(\left(\frac{1}{4}\right)^{n-1}\right)^2 = \left(\frac{1}{4}\right)^{2(n-1)} = \left(\frac{1}{16}\right)^{n-1}
     \]
   - The number of squares added in step \(n\) is \(3^{n-1} \times 4\).
   - The total area added in step \(n\) is:
     \[
     3^{n-1} \times 4 \times \left(\frac{1}{16}\right)^{n-1} = 4 \times \left(\frac{3}{16}\right)^{n-1}
     \]
   - The total area of all the squares added in all steps is the sum of an infinite geometric series with the first term \(a = 4\) and common ratio \(r = \frac{3}{16}\):
     \[
     \sum_{n=1}^{\infty} 4 \left(\frac{3}{16}\right)^{n-1} = \frac{4}{1 - \frac{3}{16}} = \frac{4}{\frac{13}{16}} = \frac{4 \times 16}{13} = \frac{64}{13}
     \]

6. **Step 6: Calculate the total area covered by all squares.**
   - The total area covered by all squares is the sum of the area of the initial square, the area of the squares added in step 2, and the sum of the infinite series:
     \[
     16 + 4 + \frac{64}{13} = \frac{208}{13} + \frac{64}{13} = \frac{272}{13}
     \]

The final answer is \( \boxed{285} \).