
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
Define an operation on $\mathbb{R}$: $p \otimes q = - \frac{1}{3}(p - c)(q - b) + 4bc$ (where $b, c \in \mathbb{R}$ are constants). Let $f_1(x) = x^2 - 2c$, $f_2(x) = x - 2b$, and $f(x) = f_1(x)f_2(x)$.
1. If the function $f(x)$ has an extremum of $- \frac{4}{3}$ at $x = 1$, determine the values of $b$ and $c$.
2. Find the common points between the curve $y = f(x)$ and the tangent lines with a slope of $c$.
3. Let $g(x) = |f'(x)|$ for $-1 \leq x \leq 1$, and denote the maximum value of $g(x)$ by $M$. If $M \geq k$ holds for any $b, c$, determine the range of values for $k$. (Reference formula: $x^3 - 3bx^2 + 4b^3 = (x + b)(x - 2b)^2$)

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