
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 that $a > 0$, let set $A = \{x \mid ax^2 - 2x + 2a - 1 = 0\}$ and set $B = \{y \mid y = \log_2(x + \frac{a}{x} - 4)\}$. Consider propositions $p$: $A = \emptyset$ and $q$: $B = \mathbb{R}$.
1. If $p \wedge q$ is true, find the maximum value of $a$.
2. If $p \wedge q$ is false, but $p \vee q$ is true, find the range of possible values for $a$.

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