
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 $f(x)=\ln (ax+1)+\frac{{{x}^{3}}}{3}-{{x}^{2}}-ax(a\in R)$.

(1) Find the range of values for the real number $a$ such that the function $y=f(x)$ is strictly increasing on the interval $[4,+\infty)$.

(2) When $a\geqslant \frac{{5}}{{2}}$, let $g(x)=\ln \[{{x}^{2}}(ax+1)\]+{{x}^{3}3}−3ax−f(x)(x > 0)$ and its two extreme points $x_{1}$ and $x_{2}(x_{1} < x_{2})$ are the roots of $φ(x)=\ln x-cx^{2}-bx$. Find the minimum value of $y=({{x}\_{1}}-{{x}\_{2}}){\varphi }{{{'}}}\\left( \\frac{{{x}\_{1}}+{{x}\_{2}}}{2} \\right)$.

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