
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
We have  $2021$  colors and  $2021$  chips of each color. We place the  $2021^2$  chips in a row. We say that a chip  $F$  is *bad* if there is an odd number of chips that have a different color to  $F$  both to the left and to the right of  $F$ .

(a) Determine the minimum possible number of bad chips.

(b) If we impose the additional condition that each chip must have at least one adjacent chip of the same color, determine the minimum possible number of bad chips.

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