
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
In the Cartesian coordinate system $(xOy)$, the parametric equations of the curve $C\_1$ are given by $\begin{cases} x=2\cos φ, \ y=\sin φ, \end{cases}$ where $(φ)$ is the parameter. In the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the curve $C\_2$ is a circle with center $(2, \frac{π}{2})$ and radius $1$.

(I) Find the Cartesian equations of the curves $C\_1$ and $C\_2$.

(II) Let $M$ be a point on the curve $C\_1$ and $N$ be a point on the curve $C\_2$. Find the range of values for the distance $|MN|$.

II. Given $a > 0$ and $b > 0$, the minimum value of the function $f(x)=|x+a|+|x-b|$ is $4$.

(I) Find the value of $a+b$.

(II) Find the minimum value of $\frac{1}{4}a^{2} + \frac{1}{4}b^{2}$.

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