
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 $f(x)= \begin{cases} a\ln x+(x-c)^{2}, & x\geqslant c \\ a\ln x-(x-c)^{2}, & 0 < x < c \end{cases}$ where $a < 0$, $c > 0$  
$(1)$ When $a=2c-2$, if $f(x)\geqslant \frac {1}{4}$ holds for any $x\in(c,+\infty)$, find the range of real numbers $a$;  
$(2)$ Suppose the graph of the function $f(x)$ has tangents at points $P(x_{1},f(x_{1}))$, $Q(x_{2},f(x_{2}))$ denoted as $l_{1}$, $l_{2}$ respectively. If $x_{1}= \sqrt {- \frac {a}{2}}$, $x_{2}=c$, and $l_{1}$ intersects $l_{2}$, find the minimum value of the real number $c$.

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