
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
Given that the origin of the rectangular coordinate system is the pole, the non-negative semi-axis of the x-axis is the polar axis, and the same length unit is taken in both coordinate systems. The parameter equation of the line $l$ is $\begin{cases} x=1+\frac{1}{2}t \\ y=\frac{\sqrt{3}}{2}t \end{cases} (t \text{ is the parameter})$. The polar coordinate equation of the curve $C$ is $(1+\sin^{2}\theta)\rho^{2}=2$.

(1) Write the ordinary equation of the line $l$ and the rectangular coordinate equation of the curve $C$;

(2) If the line $l$ intersects the curve $C$ at points $A$ and $B$, and point $P$ is $(1,0)$, find $\frac{1}{|AP|^{2}}+\frac{1}{|BP|^{2}}$.

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