
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 \( N \) be the number of distinct roots of \(\prod_{k=1}^{2012}\left(x^{k}-1\right)\). Give lower and upper bounds \( L \) and \( U \) on \( N \). If \( 0 < L \leq N \leq U \), then your score will be \(\left\lfloor\frac{23}{(U / L)^{1.7}}\right\rfloor\). Otherwise, your score will be 0.

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