
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
Is there any \\(\alpha\\) and \\(\beta\\), with \\(\alpha \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\\) and \\(\beta \in (0,\pi)\\), that satisfy the equations \\(\sin (3\pi-\alpha) = \sqrt{2}\cos \left(\frac{\pi}{2}-\beta\right)\\) and \\(\sqrt{3}\cos (-\alpha) = -\sqrt{2}\cos (\pi+\beta)\\) simultaneously? If they exist, find the values of \\(\alpha\\) and \\(\beta\\); if not, explain why.

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