
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
Transform every point on the circle $x^{2}+y^{2}=4$ by keeping the y-coordinate unchanged and doubling the x-coordinate to obtain curve $C$.

$(1)$ Find the standard equation of curve $C$;

$(2)$ Let the line $l: x-2y+4=0$ intersect curve $C$ at points $P_{1}$ and $P_{2}$. With the origin as the pole and the positive x-axis as the polar axis, establish a polar coordinate system and find the polar equation of the circle with diameter $P_{1}P_{2}$.

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