
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 diagram shows twenty congruent [circles](https://artofproblemsolving.com/wiki/index.php/Circle) arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The [ratio](https://artofproblemsolving.com/wiki/index.php/Ratio) of the longer dimension of the rectangle to the shorter dimension can be written as $\dfrac{1}{2}(\sqrt{p}-q)$ where $p$ and $q$ are positive integers. Find $p+q$.

[AIME 2002I Problem 02.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_2002I_Problem_02.png)

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