
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 left and right foci of the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ are $F_{1}$ and $F_{2}$, respectively, and its eccentricity is $\dfrac {1}{2}$. Point $P$ is a moving point on the ellipse, and the maximum area of $\triangle PF_{1}F_{2}$ is $\sqrt {3}$.  
$(1)$ Find the equation of the ellipse;  
$(2)$ If the equation of curve $C$ is $(x-t)^{2}+y^{2}=(t^{2}+2t)^{2}(0 < t\leqslant \dfrac { \sqrt {2}}{2})$, and a line $l$ passing through point $A(-2,0)$ is tangent to curve $C$, find the minimum length of the line segment cut by the ellipse on line $l$.

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