
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 $f(x) = \log_a\left( \sqrt{x^2 + 1} + x \right) + \frac{1}{a^x - 1} + 1$ ($a>0$ and $a\neq1$), if $f\left(\sin\left(\frac{\pi}{6} - \alpha\right)\right) = \frac{1}{3}$, find the value of $f\left(\cos\left(\alpha - \frac{2\pi}{3}\right)\right)$.

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