
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 curve $C_{1}$ are $\left\{\begin{array}{l}x=1+t\cos\alpha\\ y=1+t\sin\alpha\end{array}\right.$ (where $t$ is the parameter), $\alpha \in \left[0,\pi \right)$. Taking $O$ as the pole and the non-negative $x$-axis as the polar axis, establish a polar coordinate system. The polar coordinate equation of curve $C_{2}$ is $\rho =4\cos \theta$. <br/>$(1)$ Find the rectangular coordinate equation of $C_{2}$; <br/>$(2)$ Given the point $P(1,1)$, let $A$ and $B$ be the intersection points of $C_{1}$ and $C_{2}$. When $\frac{1}{{|PA|}^{2}}+\frac{1}{{|PB|}^{2}}=1$, find the polar coordinate equation of $C_{1}$.

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