
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 $f(x) = ax^2 + bx + c$ with two zeros $x\_1$, $x\_2$, and $f(1) = 2a$.
(I) Find the range of $\frac{b}{a}$;
(II) If $a > c$, and the maximum value of the function $g(x) = f(x - x\_1) + f(x - x\_2)$ in the interval $[0, 1]$ is $\frac{2}{a}$, determine whether the point $(a, b)$ is on the line $x + y = 1$? Explain your reasoning.

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