
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 function $y = \lg(-x^2 + x + 2)$ with domain $A$, find the range $B$ for the exponential function $y = a^x$ $(a>0$ and $a \neq 1)$ where $x \in A$.

1. If $a=2$, determine $A \cup B$;
2. If $A \cap B = (\frac{1}{2}, 2)$, find the value of $a$.

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