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Building a landing page with Sinatra, Google Spreadsheets, and Ruby is possible by using the Google Spreadsheet API to collect user data and store it in a spreadsheet. The process starts by setting up a new spreadsheet and service account, then creating a template project with necessary dependencies. A form is added to the index page, which collects name, email, and phone number details from users, and these parameters are sent to the Google Spreadsheets API for storage. The app can validate user input using Active Model Validations, and once data is stored in the spreadsheet, it can be manipulated with Apps Script, such as sending SMS messages.