
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 $f(x)=\ln x-ax^{2}+x$.  
$(1)$ If $f(x) \lt 0$ holds for any real number $x \in (0,+\infty)$, find the range of the real number $a$;  
$(2)$ When $a=\frac{1}{2}$, if $f(x_{1})+f(x_{2})=1$, find the minimum value of $x_{1}+x_{2}$.

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