
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 triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and the area is $S$. It is given that $6S=a^{2}\sin A+b^{2}\sin B$. Express $\cos C$ in terms of $a$, $b$, and $c$ when the maximum value of $\frac{{a+b}}{c}$ is achieved.

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