
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 the function $f(x)=\begin{cases}-x^{2}+2ax & (x\leqslant 1) \\ (2a-1)x-3a+6 & (x > 1)\end{cases}$ satisfies: for any real numbers $x_{1}\neq x_{2}$, $\left(x_{1}-x_{2}\right)\left[f\left(x_{1}\right)-f\left(x_{2}\right)\right] > 0$, determine the range of the real number $a$.

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