
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 $\triangle ABC$, the sides $a$, $b$, and $c$ opposite to the angles $A$, $B$, and $C$, respectively, satisfy $a \neq b$, and $2\sin (A-B)=a\sin A-b\sin B$.

(I) Find the length of side $c$.
(II) If the area of $\triangle ABC$ is 1 and $\tan C=2$, find the value of $a+b$.

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