
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 $xOy$, with the origin $O$ as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. It is known that the polar coordinate equation of the line $l$ is $\rho \sin \left( \theta -\frac{ \pi }{4} \right)=\sqrt{2}$, and the polar coordinate equation of the curve $C$ is $\rho {{\sin }^{2}}\theta =\cos \theta$.

1. Write down the rectangular coordinate equations of the line $l$ and the curve $C$.
2. A line parallel to $l$ passes through the moving point $P\left( {{x}_{0}},{{y}_{0}} \right)\left( y_{0}^{2}\lt {}{{x}_{0}} \right)$ and intersects the curve $C$ at points $A$ and $B$, such that $\left| PA \right|\cdot \left| PB \right|=2$. Find the closest distance from the moving point $P$ to the line $l$.

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