Below is an instruction that describes a task. Write a response that appropriately completes the request in English. Please answer in English.

### Instruction:
Alina and her best friend Lucia like to chat a lot. On a particular day, Alina sent 20 fewer messages than her friend Lucia, who sent 120 messages. The next day, Lucia sent 1/3 of the messages she sent the previous day, while Alina doubled the messages she sent on the first day. If they sent the same number of messages on the third day as the first day, what is the total number of messages they sent in those three days?

### Response:
On the first day, Alina sent 120 - 20 = <<120-20=100>>100 messages.
The total number of messages sent by Alina and Lucia on the first day is 100+120 = <<100+120=220>>220 messages
On the second day, Lucia sent 1/3 * 120 = <<1/3*120=40>>40 messages
Alina sent 100*2 = <<100*2=200>>200 messages on the second day.
The total number of messages sent by the two on the second day is 40+200 = <<40+200=240>>240 messages.
For the three days, Alina and Lucia sent 220+240+220 = <<220+240+220=680>>680 messages.
#### 680