
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
If we substitute every natural number greater than 1, as well as every fraction that is 1/2 greater than a natural number, into the expression \( y = \frac{x+9}{x-1} \), for which \( x \)-values is \( y \) an integer? For which \( x \)-values can the fraction be simplified? How many permissible \( x \)-values are there for which \( y \) is:
a) greater than 5,
b) no more than 3,
c) between 1.5 and 2.5, inclusive of these bounds?

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