
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 $y = x + \frac{a}{x}$, it has the following property: if the constant $a > 0$, then the function decreases on $(0, \sqrt{a}]$ and increases on $[\sqrt{a}, +\infty)$.

(Ⅰ) Suppose the range of the function $y = x + \frac{2^b}{x}$ (where $x > 0$) is $[6, +\infty)$, find the value of the real number $b$;

(Ⅱ) Given that $f(x) = \frac{4x^2 - 12x - 3}{2x + 1}$, where $x \in [0, 1]$, find the intervals of monotonicity and the range of the function $f(x)$;

(Ⅲ) For the function $f(x)$ from (Ⅱ) and the function $g(x) = -x - 2c$, suppose that for any $x_1 \in [0, 1]$, there always exists $x_2 \in [0, 1]$ such that $g(x_2) = f(x_1)$ holds true, find the value of the real number $c$.

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