
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 that when $x < 1$, $f(x) = (2-a)x + 1$; when $x \geq 1$, $f(x) = a^x$ ($a > 0$ and $a \neq 1$), determine the range of $a$ for which $\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} > 0$ holds for any $x_1 \neq x_2$.

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