
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
Let the function $f(x)= \frac {a}{3}x^{3}- \frac {3}{2}x^{2}+(a+1)x+1$, where $a$ is a real number.  
$(1)$ Given that the function $f(x)$ has an extremum at $x=1$, find the value of $a$;  
$(2)$ Given that the inequality $f′(x) > x^{2}-x-a+1$ holds for any $a\in(0,+\infty)$, find the range of real numbers $x$.

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