
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
In the dog dictionary the words are any sequence of letters  $A$  and  $U$  for example  $AA$ ,  $UAU$  and  $AUAU$ . For each word, your "profundity" will be the quantity of subwords we can obtain by the removal of some letters.
For each positive integer  $n$ , determine the largest "profundity" of word, in dog dictionary, can have with  $n$  letters.
Note: The word  $AAUUA$  has "profundity"  $14$  because your subwords are  $A, U, AU, AA, UU, UA, AUU, UUA, AAU, AUA, AAA, AAUU, AAUA, AUUA$ .

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