
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 $f\left(x\right)=\left\{\begin{array}{l}{(a-5)x-2, x≥2}\\{{x}^{2}-2(a+1)x+3a, x＜2}\end{array}\right.$, determine the range of real number $a$ for which $\frac{f({x}_{1})-f({x}_{2})}{{x}_{1}-{x}_{2}} \lt 0$ for any $x_{1}$, $x_{2} \in R (x_{1} \neq x_{2})$.

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