
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
Solve the equation \( x(x-1)(x-2) + (100-x)(99-x)(98-x) = 0 \).

There are several integers written on the board. Kolya replaced each integer as follows: for each even number, he wrote down its half, and for each odd number, he wrote down double the number. Could it be that the sum of the new numbers equals the sum of the original numbers if the sum of the original numbers was a) 2014; b) 2013?

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