
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
If the function $$f(x)= \begin{cases} a\cdot 2^{x-1}- \frac{1}{a} &, x\leq 1 \\ (a-2)x+ \frac{5}{3} &, x > 1 \end{cases}$$ where $a > 0$ and $a\neq 1$, and for any $x_1, x_2 \in \mathbb{R}$ with $x_1 \neq x_2$, it always holds that $$(x_1-x_2)[f(x_1)-f(x_2)] > 0,$$ then the range of values for $a$ is ______.

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