
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
Given circle $O$: $x^{2}+y^{2}=4$ and curve $C$: $y=3|x-t|$, there are two points $A(m,n)$ and $B(s,p)$ on curve $C$ ($m$, $n$, $s$, $p$ are all positive integers). The ratio of the distance from any point on circle $O$ to point $A$ and point $B$ is a fixed value $k(k > 1)$. Find $m^{s}-n^{p}=$ _____.

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