
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 ellipse $\dfrac {x^{2}}{36}+ \dfrac {y^{2}}{9}=1$ and the point $P(4,2)$, a line $l$ passes through point $P$ and intersects the ellipse at points $A$ and $B$.  
$(1)$ When the slope of line $l$ is $\dfrac {1}{2}$, find the length of segment $AB$;  
$(2)$ When point $P$ is exactly the midpoint of segment $AB$, find the equation of line $l$.

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