
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 $2 \le x \le 15$ such that $$q(x) = \begin{cases} x^2 + 1 &\quad \lfloor x \rfloor\text{ is even} \\ q(y) + (x + 1 - \lfloor x \rfloor) &\quad \text{otherwise} \end{cases}$$ where $y$ is the smallest prime factor of $\lfloor x \rfloor$. Determine the range of $q$ in interval notation.

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