
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
Given that ${\sin }^{2}{30}^{\circ }+{\sin }^{2}{90}^{\circ }+{\sin }^{2}{150}^{\circ }=\frac{3}{2}$ and ${\sin }^{2}{5}^{\circ }+{\sin }^{2}{65}^{\circ }+{\sin }^{2}{125}^{\circ }=\frac{3}{2}$, observe the pattern of these two equations and write a general proposition that holds true for any angle $\alpha$, and provide a proof.

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