
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 functions $$f(x) = \cos\left(\frac{2\pi}{3}x\right) + (a-1) \sin\left(\frac{\pi}{3}x\right) + a$$ and $$g(x)=2^{x} - x^{2},$$ if $$f[g(x)] \leq 0$$ for all $$x \in [0, 1],$$ determine the range of values for the real number $$a$$.

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