
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 functions $f(x)=x^{2}-ax-a\ln x (a\in\mathbb{R})$ and $g(x)=-x^{3}+ \frac{5}{2}x^{2}+2x-6$, find the range of $a$ such that $f(x)\geqslant$ the maximum value of $g(x)$ on $[1,4]$ for all $x\in[1,+\infty)$.

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