
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
Determine the numbers \( A, B, C, D \) such that

$$
\frac{A}{x+1}+\frac{B}{(x+1)^{2}}+\frac{C x+D}{x^{2}+x+1}=\frac{1}{(x+1)^{2} \cdot\left(x^{2}+x+1\right)}
$$

holds for every \( x \) different from \(-1\).

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