
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 a quadratic function $f(x) = ax^2 - bx + 1$.

(1) If the solution set of $f(x) < 0$ is $\left(\frac{1}{4}, \frac{1}{3}\right)$, find the values of the real numbers $a$ and $b$.

(2) If $a$ is a positive integer, $b = a + 2$, and the minimum value of the function $f(x)$ on the interval $[0, 1]$ is $-1$, find the value of $a$.

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