
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
In a certain geometric progression containing $2n$ positive terms, the product of the first term and the last term is 1000. Find the sum of the common logarithms (base 10 logarithms) of all the terms in the progression.

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