Accurate prediction of click-through rates (CTR) for cold-start entities (CSEs) within search engine ecosystems presents significant challenges. Notably, CSEs encompass novel users/items and new session search queries, each characterized by their limited interaction data and poor-quality embeddings, which collectively contribute to the complexity of CTR estimation.Existing studies predominantly address cold-start challenges in isolation, such as focusing separately on new users or new items, and lack a comprehensive framework to effectively integrate atomic ID features with group-level representations. To address these limitations, we propose GRAIN (Group Reinforced Adaptive Interaction Network), a novel framework that enhances CTR prediction across all maturity phases, namely Cold-Start, Warm-Up, and Common. GRAIN consists of three key components: 1) a Graph-based Id-to-Cluster (GIC) module that aggregates atomic ID features into cluster-level representations; 2) an ID-Cluster Cross (ICC) module that aligns ID-level and cluster-level features through contrastive learning and cross-grained interaction mechanism; 3) a lightweight auxiliary task that classifies entities into different maturity stages using a data-driven phase partitioning algorithm. Extensive experiments demonstrate GRAIN's effectiveness in improving CTR prediction accuracy across multiple maturity phases. GRAIN has been successfully deployed on the 1688 App, handling billions of daily requests.