
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 $k$ be a positive integer, and the coefficient of the fourth term in the expansion of $(1+ \frac{x}{k})^{k}$ is $\frac{1}{16}$. Consider the functions $y= \sqrt{8x-x^{2}}$ and $y= \frac{1}{4}kx$, and let $S$ be the shaded region enclosed by their graphs. Calculate the probability that the point $(x,y)$ lies within the shaded region $S$ for any $x \in [0,4]$ and $y \in [0,4]$.

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