
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 $f(x) = x^3 - ax^2 + bx + c$, its graph is curve $C$.
(1) If there exists a point $P$ on curve $C$ such that the tangent line at point $P$ is parallel to the x-axis, find the relationship between $a$ and $b$;
(2) If the function $f(x)$ can attain extreme values at $x = -1$ and $x = 3$, find the values of $a$ and $b$ at this time;
(3) Under the condition of (2), if $f(x) < 2c$ holds true for all $x \in [-2, 6]$, find the range of values for $c$.

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