
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
For a positive integer  $n$ ,  $(a_0, a_1, \cdots , a_n)$  is a  $n+1$ -tuple with integer entries. 
For all  $k=0, 1, \cdots , n$ , we denote  $b_k$  as the number of  $k$ s in  $(a_0, a_1, \cdots ,a_n)$ .
For all  $k = 0,1, \cdots , n$ , we denote  $c_k$  as the number of  $k$ s in  $(b_0, b_1, \cdots ,b_n)$ .
Find all  $(a_0, a_1, \cdots ,a_n)$  which satisfies  $a_0 = c_0$ ,  $a_1=c_1$ ,  $\cdots$ ,  $a_n=c_n$ .

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