
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
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\frac{{\sin B \sin C}}{{3 \sin A}} = \frac{{\cos A}}{a} + \frac{{\cos C}}{c}$, and the area of $\triangle ABC$ is ${S_{\triangle ABC}} = \frac{{\sqrt{3}}}{4}({{a^2} + {b^2} - {c^2}})$. <br/>$(1)$ Find the radius of the circumcircle of $\triangle ABC$; <br/>$(2)$ Find the range of values for $\frac{c}{{a+b}}$.

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