
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
The mathematical giant Euler was the first to represent a polynomial in terms of $x$ using the notation $f(x)$. For example:

$f(x) = x^2 + 3x - 5$. When $x = a$, the value of the polynomial is represented as $f(a)$. For example, when $x = -1$, the value of the polynomial $x^2 + 3x - 5$ is denoted as $f(-1) = (-1)^2 + 3 \times (-1) - 5 = -7$.

(1) Given $f(x) = -2x^2 - 3x + 1$, find the value of $f(-2)$.

(2) Given $f(x) = ax^3 + 2x^2 - ax - 6$, when $f\left(\frac{1}{2}\right) = a$, find the value of $a$.

(3) Given $f(x) = \frac{2bx + a}{3} - \frac{x - bk}{6} - 2$ ($a$, $b$ are constants), if $f(1) = 0$ for any rational number $k$, find the values of $a$ and $b$.

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