Recent segmentation models couple large language models (LLMs) with mask decoders to ground complex language expressions into masks, yet their instructions remain target-referential: they describe, constrain, or imply the region to be segmented. However, in real-world embodied interaction, human instructions are often at the intent-level, which includes the desired outcome without naming the region that enables it. To bridge this gap, we introduce SegWorld, where the model reasons about the scene through a multi-level visual chain-of-thought (CoT) before committing to a mask. Before receiving any instructions, it proactively observes the scene, describing visible objects and inferring plausible events they may support. Given an instruction, it continues the chain: from the object relevant to the intent, through the action that satisfies it, to the physical interaction site, the object part that affords the action. We formalize SegWorld as probabilistic inference, in which proactive observation supplies a linguistic scene context that improves mask prediction when instructions are given at the level of intent. We construct an intent-to-part benchmark for evaluating affordance-bearing part segmentation from high-level goals. Experiments show SegWorld matches instruction-driven baselines on target-referential instructions and improves substantially on intent-level ones.