
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
Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\begin{cases} x=2+t \\ y=kt \end{cases}$ ($t$ is the parameter), and the parametric equation of line $l_2$ is $\begin{cases} x=-2+m \\ y=\frac{m}{k} \end{cases}$ ($m$ is the parameter). Let $P$ be the intersection point of $l_1$ and $l_2$. When $k$ changes, the trajectory of $P$ is curve $C$.

(1) Write the standard equation of $C$; (2) Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis,

Let $l_3: \rho (\cos \theta +\sin \theta )-\sqrt{2}=0$, $M$ be the intersection point of $l_3$ and $C$, find the polar radius of $M$.

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