
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 $f(x)= \begin{cases} \log_{a}(x+a-1),(x > 1) \\ (2a-1)x-a,(x\leqslant 1)\end{cases}$ satisfies $\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}} > 0$ for any real numbers $x_{1}\neq x_{2}$, find the range of the real number $a$.

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