
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
\[ f(x) = \cos(\pi x), \quad g(x) = 2^x a - \frac{1}{2} \quad (a \neq 0). \]

If there exist \( x_1, x_2 \in [0,1] \) such that \( f(x_1) = g(x_2) \), determine the range of the real number \( a \).

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