
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
Let \( a, b, c \) be real numbers such that \( a \neq 0 \). Consider the parabola with equation
\[ y = ax^2 + bx + c, \]
and the lines defined by the six equations
\[
\begin{array}{lll}
y = ax + b, & y = bx + c, & y = cx + a, \\
y = bx + a, & y = cx + b, & y = ax + c.
\end{array}
\]
Suppose that the parabola intersects each of these lines in at most one point. Determine the maximum and minimum possible values of \( \frac{c}{a} \).

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