
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 two geometric progressions, each containing the same number of terms. The first term and the common ratio of the first progression are 20 and \( \frac{3}{4} \), respectively. The first term and the common ratio of the second progression are 4 and \( \frac{2}{3} \), respectively. If we multiply the terms of these progressions with the same indices, the sum of all such products is 158.75. Find the number of terms in these progressions.

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