
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 $f(x)=a\ln x+(x-c)|x-c|$, where $a < 0$ and $c > 0$.
(1) When $a=-\frac{3}{4}$ and $c=\frac{1}{4}$, find the monotonic intervals of the function $f(x)$;
(2) When $c=\frac{a}{2}+1$, if $f(x)\geqslant\frac{1}{4}$ holds for all $x\in(c,+\infty)$, find the range of values for the real number $a$;
(3) Let the graph of the function $f(x)$ have tangent lines $l_1$ and $l_2$ at points $P(x_1,f(x_1))$ and $Q(x_2,f(x_2))$, respectively. If $x_1=\sqrt{-\frac{a}{2}}$, $x_2=c$, and $l_1\perp l_2$, find the minimum value of the real number $c$.

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