
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 \( p(x) \) be defined on \( 2 \le x \le 15 \) such that 
\[ p(x) = \begin{cases} 
x + 2 & \quad \text{if } \lfloor x \rfloor \text{ is prime} \\
x^2 & \quad \text{if } \lfloor x \rfloor \text{ is a perfect square} \\
p(y) + (x + 1 - \lfloor x \rfloor) & \quad \text{otherwise}
\end{cases} \]
where \( y \) is the greatest prime factor of \( \lfloor x \rfloor \). Express the range of \( p \) in interval notation.

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