
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
Kira has  $3$  blocks with the letter  $A$ ,  $3$  blocks with the letter  $B$ , and  $3$  blocks with the letter  $C$ . She puts these  $9$  blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence  $ABCAABCBC$  the blocks with the letter A have distances  $1, 3$ , and  $4$  between one another, the blocks with the letter  $B$  have distances  $2, 4$ , and  $6$  between one another, and the blocks with the letter  $C$  have distances  $2, 4$ , and  $6$  between one another. Altogether, we got distances of  $1, 2, 3, 4$ , and  $6$ ; these are  $5$  distinct distances. What is the maximum number of distinct distances that can occur?

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