
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 $q(x)$ be defined on $3 \le x \le 15$ such that:
$$q(x) = \begin{cases} x + 2 &\quad \lfloor x \rfloor\text{ is an odd prime} \\ q(z) + (x + 2 - \lfloor x \rfloor) &\quad \text{otherwise} \end{cases}$$ 
where $z$ is the greatest prime factor of $\lfloor x\rfloor.$ Express the range of $q$ in interval notation.

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