
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 the set $M=\{1,2,3,4,5,6\}$. For $a_i, b_i \in M$, denote $e_i= \frac{b_i}{a_i}$ and $a_i < b_i$. The set formed by all $e_i$ is denoted as $A=\{e_1, e_2, \ldots, e_k\}$.
- (I) Find the value of $k$;
- (II) Let the set $B=\{e'_i | e'_i= \frac{1}{e_i}, e_i \in A\}$. For any $e_i \in A$ and $e'_j \in B$, find the value of $\sum\limits_{i \neq j}e_i - e'_j$;
- (III) Given $e_i \in A$ and $e'_j \in B$, find the probability of $e_i + e'_j \in \mathbb{Z}$.

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