
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
Find the value of the sum
$$
S = [\lg 2] + [\lg 3] + \cdots + [\lg 2008] + \left[\lg \frac{1}{2}\right] + \left[\lg \frac{1}{3}\right] + \cdots + \left[\lg \frac{1}{2008}\right]
$$
where $[x]$ denotes the greatest integer less than or equal to $x$.

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