
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, Coach $F$ of country $F$ plans to have players $A_{1}, A_{2}, \cdots, A_{7}$ participate in three practice matches (each lasting 90 minutes). Suppose at any given moment, only one of these players is on the field, and the total playing time (in minutes) for $A_{1}, A_{2}, A_{3}, A_{4}$ must be divisible by 7, while the total playing time for $A_{5}, A_{6}, A_{7}$ must be divisible by 13. If there are no restrictions on the number of substitutions, how many different possible allocations of total playing time are there, considering the conditions for each player?

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