
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
Find all numbers \(a, b, c, d\) such that:
(i) The function \(f(x)=4x^{3} - dx\) satisfies the inequality \( |f(x)| \leq 1 \) for \(x \in [-1,1]\);
(ii) The function \(g(x)=4x^{3} + ax^{2} + bx + c\) satisfies the inequality \( |g(x)| \leq 1 \) for \(x \in [-1,1]\).

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