
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 quadrilateral $ABCD$ is inscribed in a circle, and all four sides $AB, BC, CD, DA$ have positive integer lengths with $DA = 2005$. The angles $\angle ABC = \angle ADC = 90^{\circ}$, and $\max \{AB, BC, CD\} < 2005$. Find the maximum and minimum possible values of the perimeter of quadrilateral $ABCD$.

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