
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 an ellipse $C$ with a major axis length of $2\sqrt{6}$, and the coordinate of the left focus is $(-2,0)$;  
$(1)$ Find the standard equation of $C$;  
$(2)$ Suppose a line $l$, which is not perpendicular to the $x$-axis, passes through the right focus of $C$ and intersects $C$ at points $A$ and $B$, with $|AB| = \sqrt{6}$, find the slope angle of line $l$.

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