
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
There are  $2022$  signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length  $1011$  whose each term is either red or blue, Auto can always remove  $1011$  signs from the line so that the remaining  $1011$  signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.

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