
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
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$, respectively. Given that angle $A$ is obtuse and $a\sin B=b\cos B$. Find:<br/>
$(1)$ If $C=\frac{\pi}{6}$, determine the value of angle $A$.<br/>
$(2)$ Find the range of values for $\cos A+\cos B+\cos C$.

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