
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 $a, b, c, d$ be real numbers, none of which are equal to $-1$ or $-i$, and let $\omega$ be a complex number such that $\omega^3 = 1$ and $\omega \neq 1$. Let $\psi = \omega^2$ (the conjugate of $\omega$). If
$$\frac{1}{a + \psi} + \frac{1}{b + \psi} + \frac{1}{c + \psi} + \frac{1}{d + \psi} = \frac{4}{\omega},$$
then find the sum:
$$\frac{1}{a+1} + \frac{1}{b+1} + \frac{1}{c+1} + \frac{1}{d+1}.$$

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