
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
For what values of the parameter \( a \) does the equation \( |f(x) - 4| = p(x) \), where \( f(x) = \left| \frac{x^2 + 3x}{x+3} - \frac{x^2 - 4x + 4}{2-x} \right| \) and \( p(x) = a \), have three solutions? If there is more than one value for the parameter, give the product of these values.

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