
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 $h\left(x\right)$, the parametric equations of the line $l_{1}$ are $\left\{\begin{array}{l}x=4-t\\y=kt\end{array}\right.$, where $t$ is a parameter. The general equation of the line $l_{2}$ is $y=\dfrac{1}{k}x$. Let $P$ be the intersection point of $l_{1}$ and $l_{2}$. As $k$ varies, the locus of point $P$ is denoted as curve $C_{1}$. In the polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, the equation of the line $l_{3}$ is $\rho \sin \left(\theta -\dfrac{\pi }{4}\right)=\sqrt {2}$.

1. Find the general equation of curve $C_{1}$.
2. Let point $A$ lie on $l_{3}$ and point $B$ lie on $C_{1}$. If the angle between line $AB$ and $l_{3}$ is $\dfrac{\pi }{4}$, find the maximum value of $|AB|$.

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