
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
Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is 
$$
\begin{cases} 
& x=t, \\ 
& y=m-t 
\end{cases}
(t \text{ is the parameter, } m\in \mathbb{R})
$$
With the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C_2$ is 
$$
\rho^2=\frac{3}{1+2\sin^2\theta} (\rho > 0,\theta \in [0,\pi]).
$$

(Ⅰ) Find the Cartesian coordinate equations of curves $C_1$ and $C_2$.

(Ⅱ) If $P$ and $Q$ are moving points on $C_1$ and $C_2$ respectively, and the minimum distance between $P$ and $Q$ is $2\sqrt{2}$, find the value of the real number $m$.

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