
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 angles $\theta,$ $0 \le \theta \le 2 \pi,$ such that for all real numbers $x,$ $0 \le x \le 1,$
\[
x^2 \cos \theta - x(1 - x) + (1 - x)^2 \sin \theta \ge 0.
\]

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