
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 $f(x) = \frac{ax-1}{x^2-1}$,  
(Ⅰ) Solve for $x$ given that the solution set of the inequality $\frac{ax-1}{x^2-1} > 0$ is $\{a|a > \frac{1}{3}\}$.  
(Ⅱ) Solve the inequality $\frac{ax-1}{x^2-1} > 0$ for $x$ when $a \leq 0$.

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