
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
For  $a>0$ , let  $f(a)=\lim_{t\to\+0} \int_{t}^1 |ax+x\ln x|\ dx.$  Let  $a$  vary in the range  $0 <a< +\infty$ , find the minimum value of  $f(a)$ .

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