
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 a set of  $343$  closed jars, each containing blue, yellow and red marbles with the number of marbles from each color being at least  $1$  and at most  $7$ . No two jars have exactly the same contents. Initially all jars are with the caps up. To flip a jar will mean to change its position from cap-up to cap-down or vice versa. It is allowed to choose a
triple of positive integers  $(b; y; r) \in \{1; 2; ...; 7\}^3$  and flip all the jars whose number of blue, yellow and red marbles differ by not more than  $1$  from  $b, y, r$ , respectively. After  $n$  moves all the jars turned out to be with the caps down. Find the number of all possible values of  $n$ , if  $n \le 2021$ .

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