
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 equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{3}{5}t\\ y=1+\frac{4}{5}t\end{array}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis. The polar coordinate equation of curve $C$ is ${\rho^2}=\frac{2}{{1+{{\sin}^2}\theta}}$, where the polar coordinates of point $P$ are $(\sqrt{2},\frac{π}{4})$.<br/>$(1)$ Find the Cartesian equation of $C$ and the Cartesian coordinates of $P$;<br/>$(2)$ Let $l$ intersect $C$ at points $A$ and $B$, with the midpoint of line segment $AB$ being $M$. Find $|PM|$.

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