
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 equation of curve $C_1$ is $\begin{cases} x=t \\ y=m+t \end{cases}$ (where $t$ is the parameter, $m\in\mathbb{R}$). Taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $\rho^2=\frac{3}{3-2\cos^2\theta}$.

$(1)$ Find the general equation of curve $C_1$ and the Cartesian coordinate equation of curve $C_2$;

$(2)$ Given that point $P$ is on curve $C_2$, if the minimum distance from point $P$ to curve $C_1$ 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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