
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 $f(x)=\log _{a}(\sqrt{x^{2}+1}+x)+\frac{1}{a^{x}-1}+1 (a > 0, a \neq 1)$, if $f(\sin (\frac{\pi}{6}-\alpha))=\frac{1}{3}$, find $f(\cos (\alpha-\frac{2\pi}{3}))=$ \_\_\_\_\_\_.

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