
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 $f(x)= \frac {1}{3}x^{3}- \frac {1}{2}(a+4)x^{2}+(3a+5)x-(2a+2)\ln x.$  
(1) If $a < -1$, and $F(x)=f(x)- \frac {1}{3}x^{3}+ \frac {1}{2}(a+5)x^{2}-(2a+6)x$, discuss the monotonicity of the function $F(x)$;  
(2) Given $g(x)=f′(x)+ \frac {2a+2}{x}$, if the inequality $g(x)\geqslant \frac {2}{3}\ln x+3a+ \frac {14}{3}$ holds for all $x\in(0,+\infty)$, find the range of the real number $a$.

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