
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 $0 \lt x \lt \frac{1}{2}$, find the maximum value of $y=\frac{1}{2}x\left(1-2x\right)$;<br/>$(2)$ Given $x \lt 3$, find the maximum value of $f\left(x\right)=\frac{4}{{x-3}}+x$;<br/>$(3)$ Given $x$ and $y\in R^{+}$, and $x+y=4$, find the minimum value of $\frac{1}{x}+\frac{3}{y}$.

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