
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 soccer tournament, the coach of country $F$ wants to evaluate players $A_{1}, A_{2}, \cdots, A_{7}$ by having them participate in three training matches (each match lasting 90 minutes). Suppose that at any moment during the games, exactly one of these players is on the field, and the total playing time (in minutes) for $A_{1}, A_{2}, A_{3}, A_{4}$ is divisible by 7, and the total playing time (in minutes) for $A_{5}, A_{6}, $ and $A_{7}$ is divisible by 13. If there is no restriction on the number of substitutions, how many different possible distributions of total playing time are there for each player?

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