
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 that the odd function $f(x) = x^3 + bx^2 + cx + d$ is defined on $\mathbb{R}$ and attains extreme values at $x = \pm 1$.

(Ⅰ) Find the explicit formula for the function $f(x)$.
(Ⅱ) Demonstrate that for any two values of the independent variable $x_1$ and $x_2$ in the interval $[-1,1]$, the inequality $|f(x_1) - f(x_2)| \leq 4$ holds true.
(Ⅲ) If a point $P(m, n)$ (where $m, n \in \mathbb{R}$ and $|m| < 2$) allows for three tangents to the curve $y = f(x)$, determine the area of the corresponding region in the plane for point $P$.

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