
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 $p,$ $q,$ $r,$ $s$ be real numbers, none of which are equal to $-1,$ and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1.$ If
\[
\frac{1}{p + \omega} + \frac{1}{q + \omega} + \frac{1}{r + \omega} + \frac{1}{s + \omega} = \frac{3}{\omega^2},
\]
then find
\[
\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r + 1} + \frac{1}{s + 1}.
\]

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