
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 Cartesian coordinate system $xOy$, the parametric equations of the line $l_1$ are $\begin{cases}x=4-t \\ y=kt \\ \end{cases}$, where $t$ is a parameter. The standard equation of the line $l_2$ is $y=\frac{1}{k}x$. 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 $\rho \sin \left( \theta -\frac{\pi }{4} \right)=\sqrt{2}$.

1. Find the standard 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 $\frac{\pi }{4}$, find the maximum value of $|AB|$.

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