
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 plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{\sqrt{2}}{2}t,\\ y=\frac{\sqrt{2}}{2}t\end{array}\right.$ (where $t$ is the parameter). Using $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C$ is $\rho ^{2}+2\rho ^{2}\sin ^{2}\theta -3=0$. <br/>$(1)$ Find a parametric equation of the curve $C$; <br/>$(2)$ Let $A$ and $B$ be the points where $l$ intersects $C$, and $M$ be the point where $l$ intersects the $x$-axis. Find the value of $|MA|^{2}+|MB|^{2}$.

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