The purpose of a Project Brief in PRINCE2 is to provide a solid basis for the initiation of the project. The Project Brief includes the outline of the project's objectives, scope, constraints, and assumptions. It is developed during the 'Starting up a Project' process and is used to ensure that a clear, concise, and feasible plan is proposed before the project is formally initiated.
PRINCE2 7 Foundation resources and materials.

The 'Starting up a Project' process in PRINCE2is typically triggered when a project mandate is available.
The project mandate provides the basic information necessary to define whether there is sufficient justification to proceed with the project's setup.References: PRINCE2Foundation Manual

In PRINCE2, the requirements for a product should be captured early in the process, specifically when the project team realizes the need for that product. This approach ensures that the project delivers what is actually required and expected. Capturing requirements early helps in detailed planning and allows for effective quality management throughout the product's development and delivery phases.
(PRINCE2 7 Foundation Handbook, AXELOS)

The project board reviews the Project Initiation Documentation (PID) during an end stage assessment primarily to review summarized progress to date and to decide the future course of action for the project. This review helps determine whether the project is still viable and if the project should continue into its next stage based on performance against the plan and the business case.References: PRINCE2 Foundation Manual