
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
Let  $ n $  be a positive integer. A sequence  $(a_1,a_2,\ldots,a_n)$  of length is called  $balanced$  if for every  $ k $   $(1\leq k\leq n)$  the term  $ a_k $  is equal with the number of distinct numbers from the subsequence  $(a_1,a_2,\ldots,a_k).$ a) How many balanced sequences  $(a_1,a_2,\ldots,a_n)$  of length  $ n $  do exist?
b) For every positive integer  $m$  find how many balanced sequences  $(a_1,a_2,\ldots,a_n)$  of length  $ n $  exist such that  $a_n=m.$ 

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