
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) Calculate the product    $$ \left(1+\frac{1}{2}\right)   \left(1+\frac{1}{3}\right)  \left(1+\frac{1}{4}\right)...  \left(1+\frac{1}{2006}\right)  \left(1+\frac{1}{2007}\right) $$  b) Let the set   $$ A =\left\{\frac{1}{2},  \frac{1}{3},\frac{1}{4}, ...,\frac{1}{2006}, \frac{1}{2007}\right\} $$  Determine the sum of all products of  $2$ , of  $4$ , of  $6$ ,... , of  $2004$  ¸and of  $ 2006$  different elements of the set  $A$ .

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