
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
If the diagonal of a perpendicular equilibrium plane forms angles $\alpha_{1}, \alpha_{2}, \alpha_{3}$ with the side-diagonals from the common vertex, then:

$$
\sin^{2} \alpha_{1} + \sin^{2} \alpha_{2} + \sin^{2} \alpha_{3} = 1
$$

and

$$
\cos^{2} \alpha_{1} + \cos^{2} \alpha_{2} + \cos^{2} \alpha_{3} = 2
$$

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