
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 $f(x)= \frac {1}{3}x^{3}- \frac {1}{2}x^{2}+ax$, where $a\in\mathbb{R}$.  
(I) If $x=2$ is an extremum of $f(x)$, find the value of $a$, and discuss the monotonicity of $f(x)$;  
(II) Given the function $g(x)=f(x)- \frac {1}{2}ax^{2}+ \frac {2}{3}$, if $g(x)$ has a zero in the interval $(0,1)$, find the range of $a$;  
(III) Suppose $f(x)$ has two extremum points $x_{1}$, $x_{2}$, discuss whether the line passing through the points $(x_{1},f(x_{1}))$, $(x_{2},f(x_{2}))$ can pass through the point $(1,1)$. If it can, find the value of $a$; if it cannot, explain why.

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