
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
1. Given $$\cos\left(\alpha+ \frac {\pi}{6}\right)-\sin\alpha= \frac {3 \sqrt {3}}{5}$$, find the value of $$\sin\left(\alpha+ \frac {5\pi}{6}\right)$$;  
2. Given $$\sin\alpha+\sin\beta= \frac {1}{2}$$ and $$\cos\alpha+\cos\beta= \frac {\sqrt {2}}{2}$$, find the value of $$\cos(\alpha-\beta)$$.

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