
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
Suppose two dice are rolled consecutively, and the scores obtained are represented by $m$ and $n$ respectively. Define two vectors, $\overrightarrow{a}=(m,n)$ and $\overrightarrow{b}=(1,-3)$.

(I) Find the probability that event "$\overrightarrow{a} \perp \overrightarrow{b}$" occurs.

(II) Find the probability that event "$|\overrightarrow{a}| \leq |\overrightarrow{b}|$" occurs.

(III) Find the probability that the event "the line $y=\frac{m}{n}x$ intersects with the circle $(x-3)^2+y^2=1$" occurs.

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