
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
Given triangle  $ABC$ . Let  $A_1B_1$ ,  $A_2B_2$ , $ ...$ ,  $A_{2008}B_{2008}$  be  $2008$  lines parallel to  $AB$  which divide triangle  $ABC$  into  $2009$  equal areas. Calculate the value of   $$  \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor $$  

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