
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  $a_1$ ,  $a_2, \dots, a_{2015}$  be a sequence of positive integers in  $[1,100]$ .
Call a nonempty contiguous subsequence of this sequence *good* if the product of the integers in it leaves a remainder of  $1$  when divided by  $101$ .
In other words, it is a pair of integers  $(x, y)$  such that  $1 \le x \le y \le 2015$  and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \]Find the minimum possible number of good subsequences across all possible  $(a_i)$ .

*Proposed by Yang Liu*

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