
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
For any positive real number \( a \), let \( f(x) = |\lg x| \) denote the function's minimum value on \( [a, +\infty) \) as \( m_{a} \). Let \( g(x) = \sin \frac{\pi x}{2} \) denote the function's maximum value on \( [0, a] \) as \( M_{a} \). If \( M_{a} - m_{a} = \frac{1}{2} \), determine all possible values of \( a \).

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