次第【しだい】

次第 after a ます-stem means "as soon as": the moment one thing is done, the next follows. The second half is something you will do or ask someone to do, and the whole pattern is formal – announcements, business emails, staff speaking to customers.

A する-noun can drop the し: とうちゃくしだい到着次第, けってい決定しだい次第.

What the second half can be

次第 looks forward, so the second half is an intention, a promise or a request. It cannot report what happened, and it cannot be something nobody controls, such as the weather. For those, use たらすぐ or とすぐ.

• × 駅に着き次第、雨が降り出した → 駅に着いたらすぐ、雨が降り出した
• × 家に帰り次第、電話しました → 家に帰ってすぐ、電話しました

In conversation, たらすぐ does the same job as 次第: ついたら着いたらすぐでんわ電話するね.

Depending on

After a noun, 次第 means "depending on" or "is up to": the noun decides how things turn out. That use has its own grammar point under しだい次第だ and しだい次第で.