
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
If the function $f(x)$ satisfies $f(x)+f(2a-x)=2b$ for any $x$ in its domain, then the graph of the function $f(x)$ is symmetric about the point $(a,b)$.  
$(1)$ Given that the graph of the function $f(x)= \frac {x^{2}+mx+m}{x}$ is symmetric about the point $(0,1)$, find the value of the real number $m$;  
$(2)$ Given that the graph of the function $g(x)$ on $(-\infty,0)\cup(0,+\infty)$ is symmetric about the point $(0,1)$, and when $x\in(0,+\infty)$, $g(x)=x^{2}+ax+1$, find the expression of the function $g(x)$ on $(-\infty,0)$;  
$(3)$ Under the conditions of $(1)$ and $(2)$, if for real numbers $x < 0$ and $t > 0$, it always holds that $g(x) < f(t)$, find the range of the real number $a$.

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