
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
The following diagram shows four adjacent  $2\times 2$  squares labeled  $1, 2, 3$ , and  $4$ . A line passing through the lower left vertex of square  $1$  divides the combined areas of squares  $1, 3$ , and  $4$  in half so that the shaded region has area  $6$ . The difference between the areas of the shaded region within square  $4$  and the shaded region within square  $1$  is  $\frac{p}{q}$  , where  $p$  and  $q$  are relatively prime positive integers. Find  $p + q$ .
![Image](https://cdn.artofproblemsolving.com/attachments/7/4/b9554ccd782af15c680824a1fbef278a4f736b.png)

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