
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 $x,$ $y,$ $z$ be real numbers, none of which are equal to $-1,$ and let $\omega$ be a complex number such that $\omega^3 = 1$ and $\omega \neq 1.$ If
\[\frac{1}{x + \omega} + \frac{1}{y + \omega} + \frac{1}{z + \omega} = \omega,\]then find
\[\frac{1}{x + 1} + \frac{1}{y + 1} + \frac{1}{z + 1}.\]

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