
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
Twelve balls are numbered by the numbers  $1,2,3,\cdots,12$ . Each ball is colored either red or green, so that the following two conditions are satisfied:

(i) If two balls marked by different numbers  $a$  and  $b$  are colored red and  $a+b<13$ , then the ball marked by the number  $a+b$  is colored red, too.

(ii) If two balls marked by different numbers  $a$  and  $b$  are colored green and  $a+b<13$ , then the ball marked by the number  $a+b$  is also colored green.

How many ways are there of coloring the balls?

Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it  :ninja: )

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