
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
Before the World Cup, the coach of country \( F \) plans to evaluate the seven players \( A_{1}, A_{2}, \cdots, A_{7} \) by ensuring they all participate in three practice matches (each lasting 90 minutes). Assuming that at any moment during the matches, exactly one of these players is on the field, and that the total playing time (in minutes) for each of \( A_{1}, A_{2}, A_{3}, A_{4} \) is divisible by 7, while the total playing time for each of \( A_{5}, A_{6}, A_{7} \) is divisible by 13, calculate the total number of different possible distributions of playing times for each player.

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