
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 $f(x)=xe^{2x}-\ln x-ax$.
$(1)$ When $a=0$, find the minimum value of the function $f(x)$ on the interval $\left[ \frac{1}{2},1 \right]$;
$(2)$ If for all $x > 0$, the inequality $f(x) \geqslant 1$ always holds, find the range of values for $a$;
$(3)$ If for all $x > 0$, the inequality $f\left( \frac{1}{x}\right)-1 \geqslant \frac{1}{x}e^{ \frac{2}{x}}+ \frac{ \frac{1}{e-1}+ \frac{1}{x}}{e^{ \frac{x}{e}}}$ always holds, find the range of values for $a$.

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