
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  $ 0 \le \alpha, \beta, \gamma \le \frac{\pi}{2}$  satisfy the conditions

 $ \sin \alpha\plus{}\sin \beta\plus{}\sin \gamma\equal{}1, \sin \alpha \cos 2\alpha\plus{}\sin \beta\cos 2\beta\plus{}\sin \gamma \cos 2\gamma\equal{}\minus{}1.$ 

Find all possible values of  $ \sin^2 \alpha\plus{}\sin^2 \beta\plus{}\sin^2 \gamma$ .

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