
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
[Hexagon](https://artofproblemsolving.com/wiki/index.php/Hexagon) $ABCDEF$ is divided into five [rhombuses](https://artofproblemsolving.com/wiki/index.php/Rhombus), $\mathcal{P, Q, R, S,}$ and $\mathcal{T,}$ as shown. Rhombuses $\mathcal{P, Q, R,}$ and $\mathcal{S}$ are [ congruent](https://artofproblemsolving.com/wiki/index.php/Congruent_(geometry)), and each has [area](https://artofproblemsolving.com/wiki/index.php/Area) $\sqrt{2006}.$ Let $K$ be the area of rhombus $\mathcal{T}$.  Given that $K$ is a [positive integer](https://artofproblemsolving.com/wiki/index.php/Positive_integer), find the number of possible values for $K$.

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