
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 $xOy$, it is known that the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}x=2t\\ y={t}^{2}\end{array}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the non-negative $x$-axis as the polar axis. The polar coordinate equation of curve $C_{2}$ is $\rho \sin \theta +\rho \cos \theta =5$. Let $A$ and $B$ be the two points where curves $C_{1}$ and $C_{2}$ intersect.<br/>$(1)$ Find the Cartesian equation of curve $C_{1}$ and the rectangular coordinate equation of curve $C_{2}$;<br/>$(2)$ Given the point $P(2,3)$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.

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