
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 line $l: y= \sqrt {3}x-2 \sqrt {3}$ passes through the right focus $F_{2}$ of the ellipse $C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a > b > 0)$, and the center of the ellipse $C$ is symmetric about the line $l$ and lies on the line $x= \frac {a^{2}}{c}$ (where $2c$ is the focal length). The line $m$ passes through the left focus $F_{1}$ of the ellipse and intersects the ellipse $C$ at points $M$ and $N$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Let $\overrightarrow{OM}\cdot \overrightarrow{ON}= \frac {2\lambda}{\tan \angle MON}\neq 0$ ($O$ is the origin), when the line $m$ rotates around point $F_{1}$, find the maximum value of $\lambda$.

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