
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
The students of classes $\boldsymbol{A}$ and $\boldsymbol{B}$. There are two classes in the school: $A$ and $B$. The students of class $A$ boast that they are taller than the students of class $B$, while the students of class $B$ are considered better mathematicians. One day, when a student from class $A$ looked down on a student from class $B$, the latter asked: what exactly does it mean that you are taller than us? Does it mean that:
1) Any one of you is taller than any one of us?
2) The tallest of you is taller than the tallest of us?
3) For any student in class $A$, there is a student in class $B$ who is shorter?
4) Every student in class $B$ is shorter than at least one student in class $A$?
5) For each student in class $A$, can a different student in class $B$ be found who is shorter?
6) For each student in class $B$, can a different student in class $A$ be found who is taller?
7) The shortest student in class $B$ is shorter than the shortest student in class $A$?
8) The number of students in class $B$ who are shorter than the shortest student in class $A$ is greater than the number of students in class $A$ who are shorter than the tallest student in class $B$?
9) The total height of the students in class $A$ is greater than the total height of the students in class $B$?
10) The average height of the students in class $A$ is greater than the average height of the students in class $B$?
11) Among you, there are more students taller than any of us, than among us, there are students taller than any of you?
12) Among you, there are more students taller than our average height, than among us, there are students taller than your average height?
13) The median height of you is taller than the median height of us (in the case that there is an even number of students in the class, the median height is considered to be the arithmetic mean of the heights of the middle pair of students)?

Overwhelmed by the barrage of questions, the student from class $A$ seemed to shrink. We ask the readers: do these questions depend on each other, and if they do, indicate which ones? In other words, identify the pairs of questions for which a positive answer to one implies a positive answer to the other. Are there equivalent questions, i.e., are there pairs of questions for which the answers must necessarily be the same? Are there pairs of questions that are dependent but not equivalent?

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