
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 $f(x)=\left\{{\begin{array}{l}{(2a-3)x-1, x<1}\\{x^2+1, x\geq1}\end{array}}\right.$, and for any $x_{1}\neq x_{2}$, it satisfies the condition $\frac{{f({{x_1}})-f({{x_2}})}}{{{x_1}-{x_2}}} > 0$, determine the range of real number $a$.

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