
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 $f(x)= \begin{cases} \frac{1}{4^{x}},x \in \left[ 0, \frac{1}{2} \right] \\ -x+1,x \in \left( \frac{1}{2}, 1 \right] \\ \end{cases}$ and $g(x)=a \sin \frac{\pi}{6}x - a + 2 (a > 0)$, if there exist $x_{1}$ and $x_{2} \in [0, 1]$ such that $f(x_{1}) = g(x_{2})$ holds true, then the range of values for the real number $a$ is _____.

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