
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
Let $p$: $f(x) = 1 + ax$, where $f(x) \geq 0$ always holds on the interval $(0, 2]$, and let $q$ be the function $g(x) = ax - \frac{a}{x} + 2\ln x$ which has an extremum in its domain.  
(1) If $p$ is true, find the range of the real number $a$;  
(2) If "p ∨ q" is true and "p ∧ q" is false, find the range of the real number $a$.

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