
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 $g(x)=\sum_{k=1}^{8}(\lfloor 2kx \rfloor -2k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. Determine the number of distinct values $g(x)$ assumes for $x \ge 0$.
A) 28
B) 30
C) 32
D) 34
E) 36

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