
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 $0 \lt \alpha \lt \beta \lt 2\pi$, and the function $f\left(x\right)=5\sin \left(x-\frac{\pi}{6}\right)$. If $f\left(\alpha\right)=f\left(\beta\right)=1$, then calculate the value of $\cos \left(\beta-\alpha\right)$.

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