
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 the function $f(x)= \begin{cases} \frac{4}{\pi} \sqrt{1-x^{2}}, & 0\leqslant x < 1 \\ 5x^{4}+1, & 1\leqslant x\leqslant 2 \end{cases}$, and the sequence ${a_{n}}$ satisfies: $a_{1}= \int_{0}^{2}f(x)dx$, $a_{n+1}-a_{n}=2n$, then the minimum value of $\frac{a_{n}}{n}$ is \_\_\_\_\_\_.

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