
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 $f(x)= \frac {1}{2}x^{2}-2\ln x+a(a\in\mathbb{R})$, $g(x)=-x^{2}+3x-4$.  
$(1)$ Find the intervals of monotonicity for $f(x)$;  
$(2)$ Let $a=0$, the line $x=t$ intersects the graphs of $f(x)$ and $g(x)$ at points $M$ and $N$ respectively. When $|MN|$ reaches its minimum value, find the value of $t$;  
$(3)$ If for any $x\in(m,n)$ (where $n-m\geqslant 1$), the graphs of the two functions are on opposite sides of the line $l$: $x-y+s=0$ (without intersecting line $l$), then these two functions are said to have an "EN channel". Investigate whether $f(x)$ and $g(x)$ have an "EN channel", and if so, find the range of $x$; if not, please explain why.

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