
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 the parametric equation of line $l$ as $\begin{cases} x= \sqrt {3}+t\cos \theta \\ y=t\sin \theta \end{cases}(t\text{ is a parameter},0\leqslant \theta < \pi)$, and the polar equation of curve $C$ as $\rho^{2}= \frac {4}{1+3\sin ^{2}\theta }$  
$(1)$ Write the standard equation of curve $C$;  
$(2)$ If $F_{1}$ is the left focus of curve $C$, and line $l$ intersects curve $C$ at points $A$ and $B$, find the minimum value of $|F_{1}A|\cdot|F_{1}B|$.

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