
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 $f(x)=-x^{3}+x^{2}+b$, $g(x)=a\ln x$.

(Ⅰ) If the maximum value of $f(x)$ on the interval $x\in\left[-\frac{1}{2},1\right)$ is $\frac{3}{8}$, find the value of the real number $b$;

(Ⅱ) If for any $x\in[1,e]$, $g(x)\geqslant -x^{2}+(a+2)x$ always holds, find the range of the real number $a$.

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