
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
Let \(\mathrm{P}\) be the set \(\{1, 2, 3, \ldots, 14, 15\}\). Consider \(\mathrm{A} = \{a_{1}, a_{2}, a_{3}\}\), a subset of \(\mathrm{P}\) where \(a_{1} < a_{2} < a_{3}\) and \(a_{1} + 6 \leq a_{2} + 3 \leq a_{3}\). How many such subsets are there in \(\mathrm{P}\)?

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