
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
Given that the polar equation of curve $C$ is $\frac{2}{{\rho}^{2}}=1+\sin ^{2}\theta $, and the parametric equation of line $l$ is $\left\{\begin{array}{l}{x=-\frac{\sqrt{2}}{2}t+1}\\{y=\frac{\sqrt{2}}{2}t}\end{array}\right.$ (where $t$ is a parameter), find:

$(1)$ Convert the polar equation of curve $C$ into rectangular coordinates.

$(2)$ Let $P$ be the intersection point of line $l$ and the $x$-axis. Line $l$ intersects curve $C$ at points $M$ and $N$. Find the value of $\frac{1}{|PM|}+\frac{1}{|PN|}$.

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