
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
If \( X=\sqrt{(100)(102)(103)(105)+(Q-3)} \) is an integer and \( R \) is the units digit of \( X \), find the value of \( R \).

If \( S \) is the sum of the last 3 digits (hundreds, tens, units) of the product of the positive roots of \( \sqrt{2012} \cdot x^{\log_{2012} x}=x^{R} \), find the value of \( S \).

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