
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
7. What is the minimum value of the product   $$ \prod_{i=1}^6\frac{a_i-a_{i+1}}{a_{i+2}-a_{i+3}} $$   given that  $(a_1, a_2, a_3, a_4, a_5, a_6)$  is a permutation of  $(1, 2, 3, 4, 5, 6)$ ? (note  $a_7 = a_1, a_8 = a_2 \ldots$ )

8. Danielle picks a positive integer  $1 \le n \le 2016$  uniformly at random. What is the probability that  $\text{gcd}(n, 2015) = 1$ ?

9. How many  $3$ -element subsets of the set  $\{1, 2, 3, . . . , 19\}$  have sum of elements divisible by  $4$ ?


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