
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
In a certain lottery, there are 60 distinct numbers, and 6 of them are drawn without replacement. Each ticket contains 6 distinct numbers from the possible 60. The highest prize, known as the "gol-no-ângulo" (perfect shot), is awarded to the player whose ticket matches all 6 drawn numbers. There is also a prize called "bola-na-trave" (near miss), where the smallest number on the ticket does not differ by more than 1 (in absolute value) from the smallest drawn number; the second smallest number on the ticket does not differ by more than 1 (in absolute value) from the second smallest drawn number, and so on up to the sixth smallest number. For example, if the winning ticket is \(\{4,7,25,48,51,60\}\), then the tickets \(\{3,6,25,49,50,59\}\) and \(\{5,6,25,47,50,60\}\) are considered "bola-na-trave" tickets, but the ticket \(\{3,4,6,24,47,50\}\) is not. Note that a "gol-no-ângulo" ticket cannot be a "bola-na-trave" ticket. Consider each ticket as a sequence of 6 numbers written in ascending order for the following items.

(a) Provide an example of a set of 6 numbers forming a "gol-no-ângulo" ticket with the smallest possible number of associated "bola-na-trave" tickets. How many possible "bola-na-trave" tickets would there be for these 6 numbers?

(b) Provide an example of a set of 6 numbers forming a "gol-no-ângulo" ticket that results in the maximum possible number of "bola-na-trave" tickets. How many possible "bola-na-trave" tickets would there be in this case?

(c) Given the drawn numbers \(\{2,3,8,11,14,17\}\), how many "bola-na-trave" tickets are associated with it?

(d) Suppose the drawn numbers are \(\{8,10,12,14,16,18\}\). In this case, how many possible "bola-na-trave" tickets are associated with it?

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