
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 the function \( f(x) = x(1 + x)^2, \, x \in (-\infty, 0] \).
(1) Find the extreme points of \( f(x) \);
(2) For any \( a < 0 \), let \( F(a) \) denote the minimum value point of \( f(x) \) on \([a, 0]\), find the minimum value of \( k = \frac{F(a)}{a} \).

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