
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 plane vector $\overrightarrow{a_m}=(m, 1)$, $\overrightarrow{b_n}=(2, n)$, where $m, n \in \{1,2,3,4\}$.

    $(I)$ List all possible outcomes of the ordered pair $(m,n)$;

    $(II)$ Let event $A$ be "the ordered pair $(m,n)$ such that $\overrightarrow{a_m} \bot (\overrightarrow{a_m}-\overrightarrow{b_n})$ holds". Calculate the probability of event $A$ occurring.

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