
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 $f(x)=|x-a|-\ln x$, $(x > 0)$, $h(x)=ax-1(a\in\mathbb{R})$

$(1)$ If $a=1$, find the monotonic intervals of $f(x)$ and the minimum value of $f(x)$;  
$(2)$ If $a > 0$, find the monotonic intervals of $f(x)$;  
$(3)$ If $\frac {\ln 2^{2}}{2^{2}}+ \frac {\ln 3^{2}}{3^{2}}+\ldots+ \frac {\ln n^{2}}{n^{2}} < \frac {h(n)(2n+1)}{2(n+1)}$, find the minimum positive integer value of $a$.

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