
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 $f(x) = tx$, where $x \in \mathbb{R}$.

(1) If $t = ax + b$, where $a, b \in \mathbb{R}$, and $-1 \leq f(-1) \leq 2$, $2 \leq f(1) \leq 4$, find the area of the plane region represented by the set of points $(a, b)$.

(2) If $t = 2 + \frac{1}{x^2 - x}$, ($x < 1$ and $x \neq 0$), find the maximum value of the function $f(x)$.

(3) If $t = x - a - 3$ ($a \in \mathbb{R}$), and the solution set of the inequality $b^2 + c^2 - bc - 3b - 1 \leq f(x) \leq a + 4$ ($b, c \in \mathbb{R}$) is $[-1, 5]$, find the values of $b$ and $c$.

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