
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
Given the function $f(x)=[x]\cdot\langle x \rangle$ and $g(x)=2x-[x]-2$, determine the lengths of the solution sets for the inequalities $f(x) > g(x)$, $f(x) = g(x)$, and $f(x) < g(x)$ for $0 \leq x \leq 2012$.

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