How is a trade deficit even possible, given the need to exchange currency?

Scenario 1: Alice and Bob are American. Li and Min are Chinese. America produces beef; China produces textiles. Alice is willing to pay $20 for a sweater. Li is willing to sell a sweater for ¥100. Min is willing to pay ¥100 for a beef steak. Bob is willing to sell a beef steak for $20. In this case, Alice and Min... show more Scenario 1:
Alice and Bob are American. Li and Min are Chinese. America produces beef; China produces textiles.
Alice is willing to pay $20 for a sweater. Li is willing to sell a sweater for ¥100.
Min is willing to pay ¥100 for a beef steak. Bob is willing to sell a beef steak for $20.

In this case, Alice and Min trade $20 for ¥100, and both are able to make their purchases. Two products of equal value are exchanged between America and China.

Scenario 2:
Alice is willing to pay $20 for a sweater. Li is willing to sell a sweater for ¥100.
Min is willing to pay ¥10 for a hamburger. Bob is willing to sell a hamburger for $2.

In this case, trade cannot occur.

Maybe Alice and Min trade $2 for ¥10, and Alice buys a pair of socks instead. Or maybe Alice trades currency with 10 people like Min who buy 10 hamburgers. The products exchanged between the two countries are still of equal value.

Alice does not have the currency to buy a ¥100 sweater unless there is demand for ¥100 worth of hamburgers.
Altering the exchange rate doesn't help. The amount of yuan that Min is willing to pay is less than the amount of yuan that Lin is willing to accept.

(Obviously in real life there are more than two countries. But it still must be the case that the total exports in a currency equal total imports in that currency.)
Update: It seems like the result would be different prices in the two countries. An American hamburger in China would have to cost as much as a sweater; a Chinese sweater in America would have to be as cheap as a hamburger. But there's no incentive for Li to export sweaters if he can't make a profit on them.
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