
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
Suppose the function $f(x) = ax + \frac{x}{x-1}$ where $x > 1$.
(1) If $a > 0$, find the minimum value of the function $f(x)$.
(2) If $a$ is chosen from the set \{1, 2, 3\} and $b$ is chosen from the set \{2, 3, 4, 5\}, find the probability that $f(x) > b$ always holds true.

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