
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 the derivative of $f(x)$ is $g(x)$, and $g(x)=\ln x-2ax+2a$, $a∈R$.

(1) Find the monotonic intervals of $g(x)$;

(2) When $a\leqslant \dfrac{1}{4}$, find the extreme points of the function $f(x)$ on $(0,2)$;

(3) When $0 < a\leqslant \dfrac{1}{4}$, if $\forall {{x}\_{1}}$, ${x}\_{2}∈(0,2)$, and ${x}\_{1}\neq x\_{2}$, we have $|g({x}\_{1})-g({x}\_{2})| < 2a| \dfrac{1}{{x}\_{1}}- \dfrac{1}{{x}\_{2}}|$, find the value of the real number $a$.

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