
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
$(1)$ Given $x＜\frac{5}{4}$, find the maximum value of $y=4x-2+\frac{1}{{4x-5}}$;<br/>$(2)$ Given $0＜x＜\frac{1}{2}$, find the maximum value of $y=\frac{1}{2}x({1-2x})$;<br/>$(3)$ Given $x \gt 0$, find the maximum value of $y=\frac{{2x}}{{{x^2}+1}}$.

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