
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 rectangular coordinate system $xOy$, the parametric equations of the curve $C_1$ are $$\begin{cases} x=1+2t \\ y=-2+t \end{cases}$$ (where $t$ is the parameter), and the polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of the curve $C_2$ is $\rho^2 = \frac{4}{1 + 3\sin^2{\theta}}$.
(1) Find the ordinary equation of the curve $C_1$ and the rectangular coordinate equation of the curve $C_2$;
(2) Suppose the curve $C_2$ undergoes a scaling transformation $$\begin{cases} x'=2x \\ y'=y \end{cases}$$ to obtain the curve $C_3$. $M(x, y)$ is any point on the curve $C_3$. Find the maximum value of the distance from point $M$ to the curve $C_1$.

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