
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 $a \gt 0$, $b \gt 0$, and $4a+b=1$, find the maximum value of $ab$;<br/>$(2)$ If positive numbers $x$ and $y$ satisfy $x+3y=5xy$, find the minimum value of $3x+4y$;<br/>$(3)$ Given $x \lt \frac{5}{4}$, find the maximum value of $f\left(x\right)=4x-2+\frac{1}{{4x-5}}$.

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