
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 polar coordinate system, the polar coordinate equation of the curve $C_1$ is $\rho = \frac{24}{4\cos \theta + 3\sin \theta}$. In the rectangular coordinate system $xOy$ with the pole as the origin $O$ and the polar axis as the positive semi-axis of the $x$-axis (both coordinate systems have the same unit length), the parametric equations of the curve $C_2$ are: $\begin{cases} x = \cos \theta \\ y = \sin \theta \end{cases}$ ($\theta$ is the parameter).

1. Find the rectangular coordinate equation of the curve $C_1$ and the ordinary equation of the curve $C_2$.
2. After the curve $C_2$ undergoes an expansion transformation $\begin{cases} x' = 2\sqrt{2}x \\ y' = 2y \end{cases}$, it becomes the curve $C_3$. If $M$ and $N$ are moving points on the curves $C_1$ and $C_3$ respectively, find the minimum value of $|MN|$.

II. Given $f(x) = |2x-a| - |x+1| (a \in \mathbb{R})$.

1. Solve the inequality $f(x) > 2$ when $a = 1$.
2. Find the range of the real number $a$ such that the inequality $f(x) + |x+1| + x > a^2 - \frac{1}{2}$ holds for all $x \in \mathbb{R}$.

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