
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 $f(x)=\ln (ax+1)+ \frac {x^{3}}{3}-x^{2}-ax(a∈R)$,
(1) Find the range of values for the real number $a$ such that $y=f(x)$ is an increasing function on $[4,+∞)$;
(2) When $a\geqslant \frac {3 \sqrt {2}}{2}$, let $g(x)=\ln [x^{2}(ax+1)]+ \frac {x^{3}}{3}-3ax-f(x)(x > 0)$ and its two extreme points $x_{1}$, $x_{2}(x_{1} < x_{2})$ are exactly the zeros of $φ(x)=\ln x-cx^{2}-bx$, find the minimum value of $y=(x_{1}-x_{2})φ′( \frac {x_{1}+x_{2}}{2})$.

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