
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
A diagonal of a regular 2006-gon \( P \) is called good if its endpoints divide the boundary of \( P \) into two parts, each containing an odd number of sides. The sides of \( P \) are also called good. Let \( P \) be divided into triangles by 2003 diagonals, none of which have common interior points. What is the greatest number of isosceles triangles, each of which has two good sides, that such a division can have?

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