
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 functions $f(x) = ax^3 + x$ and $g(x) = x^2 + px + q$,
(Ⅰ) If the function $f(x)$ has an extreme value at $x = 1$, find the value of the real number $a$.
(Ⅱ) Under the condition of (Ⅰ), the graph of $F(x) = f'(x)g(x)$ (where $f'(x)$ is the derivative of $f(x)$) is symmetric about the line $x = -1$. Determine the intervals on which $F(x)$ is monotonic.
(Ⅲ) Given the conditions in (Ⅱ), if for any $x \geq 1$, the inequality $g(x) \geq (6+\lambda)x - \lambda\ln x + 3$ always holds, find the range of values for the real number $\lambda$.

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