Choose:
a. The first time, / b. At first, / c. For the first time, ... when I came to Japan, I was lonely.
日本語訳
Twitterより。「日本に来た当初は寂しかった」の冒頭に The first time / At first / For the first time のどれを入れるか選んでください。proximity は「近さ・距離」を意味します。
The best answer is 'b.', because this phrase means something experienced in an initial instance--the conditions around which continued for a time; or it means something was done that was not done before, and it was done additional times after.
'C' means something was done and/or experienced causing certain feelings and results, but that those feelings and/results changed thereafter--and that this difference will be explained shortly.
'A', though not wrong, represents a kind of phrase--because of the comma and the use of 'when' after it--that presents an unnatural (or non-fluid break in the ) flow of the sentence, specific to spoken English, which requires context and intonation to be fully understood without confusion; and this phrase is adverbial, however is followed by an adverb. Placing them side by side is a possible challenge to the non-contextual expression of this construction (say in written English)--unless understood as dialog, with quotation marks. So this (while often done) is not the best answer for a grammatically and style-correct version of the idea being expressed. Basically, it is only acceptable in spoken English or idiomatic/dialect--in dialog.
*Proximity means distance between or from.
解答・解説の日本語訳
最適なのは b. At first。ある状態が初めの一定期間続いたことを表します。For the first time は「それまで一度もなかったことを初めて経験した」という意味合いになり、その後の変化につなげる場合があります。The first time, when ... は、話し言葉でイントネーションや文脈があれば理解できても、書き言葉では区切りが不自然で、ここでは最良の選択ではありません。