Read each pattern and its example, then write your own sentence on the line.
〜のもとで N + のもとで（のもとに）
States the conditions, rules or supervision under which something is done: 'under X'. Formal and largely written; のもとに is a close variant.
先生の指導のもとで、研究を進めている。 I'm carrying out my research under my professor's guidance.
〜末に V-た / N + の + 末（すえ）に
Says that after a long process of effort or hesitation, a final result was reached. The outcome may be good or bad. Formal and mostly written.
長い話し合いの末に、結論が出た。 After long discussions, a conclusion was finally reached.
〜あげく V-た / N + の + あげく（に）
Like 〜末に but the ending is always unwelcome: after all that trouble, a bad result came. Carries the speaker's exasperation; used in speech and writing alike.
さんざん悩んだあげく、会社を辞めてしまった。 After agonizing over it, he ended up quitting the company.
〜たところ V-た + ところ（後件は過去の事実）
Reports what was discovered as a result of doing something: 'when I did X, (it turned out that) Y'. The result is factual, not planned. Formal, common in reports and email.
店に問い合わせたところ、まだ在庫があった。 When I asked the shop, they still had it in stock.
〜か〜ないかのうちに V-dict + か + V-ない + かのうちに（後件は過去形）
Two events happen almost at the same instant: 'no sooner had X happened than Y'. Describes actual past events, so the second clause is normally past tense.
家を出るか出ないかのうちに、雨が降り出した。 No sooner had I left the house than it started to rain.
〜てはじめて V-て + はじめて（後件は気づき・理解が多い）
Says a realization or ability came only after experiencing something: 'it was only after X that Y'. Often used for lessons learned; neutral to slightly formal.
病気になってはじめて、健康のありがたさが分かった。 Only after getting sick did I understand the value of good health.