
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 that $2014=\alpha_{k}\cdot5^{k}+\alpha_{k-1}\cdot5^{k-1}+\ldots+a_{1}\cdot5^{1}+a_{0}\cdot5^{0}$, where $a_{k}$, $a_{k-1}$, $\ldots$, $a_{0}\in\mathbb{N}$, $0 < a_{k} < 5$, $0\leqslant a_{k-1}$, $a_{k-2}$, $\ldots$, $a_{1}$, $a_{0} < 5$, calculate the probability that a point, randomly selected from the set $\{a_{0}, a_{1}, \ldots, a_{k}\}$, falls inside the ellipse $\dfrac{x^{2}}{16} + \dfrac{y^{2}}{9} =1$.

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