<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Alexis has  $2020$  paintings, the  $i$ th one of which is a  $1\times i$  rectangle for  $i = 1, 2, \ldots, 2020$ . Compute the smallest integer  $n$  for which they can place all of the paintings onto an  $n\times n$  mahogany table without overlapping or hanging off the table. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
1430
Alexis has  $2020$  paintings, the  $i$ th one of which is a  $1\times i$  rectangle for  $i = 1, 2, \ldots, 2020$ . Compute the smallest integer  $n$  for which they can place all of the paintings onto an  $n\times n$  mahogany table without overlapping or hanging off the table.
test