
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $p + q$ .
![Image](https://cdn.artofproblemsolving.com/attachments/2/1/d963460373b56906e93c4be73bc6a15e15d0d6.png)

Present the answer in LaTex format: \boxed{Your answer}
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