
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
The parametric equation of line $l$ is $\begin{cases} x=1+ \frac {t}{2} \\ y= \frac { \sqrt {3}}{2}t\end{cases}$, and the polar equation of curve $C$ is $(1+\sin ^{2}\theta)\rho^{2}=2$.  
$(1)$ Write the standard equation of line $l$ and the Cartesian coordinate equation of curve $C$;  
$(2)$ Suppose line $l$ intersects curve $C$ at points $A$ and $B$. If point $P$ is at $(1,0)$, find $\left( \frac {1}{|AP|^{2}}+ \frac {1}{|BP|^{2}}\right)$.

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