
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 the function $$f(x)= \begin{cases} x^{2}+x+1, & \text{for } x\geq 0 \\ 2x+1, & \text{for } x<0 \end{cases},$$ evaluate the expression \[\cos(\alpha-\beta)+\cos(\beta-\gamma)\] given that $f(\sin\alpha+\sin\beta+\sin\gamma-1)=-1$ and $f(\cos\alpha+\cos\beta+\cos\gamma+1)=3$.

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