
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
Consider the function $f(x) = x - \frac{2}{x}$, where $x \in [1,2]$, and the function $g(x) = a\cos\frac{\pi x}{2} + 5 - 2a$, with $a > 0$. For any $x_{1} \in [1,2]$, there always exists an $x_{2} \in [0,1]$ such that $g(x_{2}) = f(x_{1})$. Determine the range of values for $a$.

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