
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 \( \theta \), \( 0 \le \theta \le 2\pi \), with the following property: For all real numbers \( x \), \( 0 \le x \le 1 \),
\[ x^3 \sin \theta + x^2 \cos \theta - x(1-x) + (1-x)^2 \sin \theta > 0. \]

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