
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 points $A(5,2)$, $B(0,-\sqrt{3}-4)$, $C(-1,-4)$,

$(1)$ Find the equation of the line passing through point $A$ and having a $y$-intercept that is twice its $x$-intercept.

$(2)$ Find the range of values for the slope angle of the line passing through point $C$ and intersecting with line segment $AB$.

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