The convergence of generative AI and web search is reshaping problem-solving for programmers. However, the lack of understanding regarding their interplay in the information-seeking process often leads programmers to perceive them as alternatives rather than complementary tools. To analyze this interaction and explore their synergy, we conducted an interview study with eight experienced programmers. Drawing from the results and literature, we have identified three major challenges and proposed three decision-making stages, each with its own relevant factors. Additionally, we present a comprehensive process model that captures programmers' interaction patterns. This model encompasses decision-making stages, the information-foraging loop, and cognitive activities during system interaction, offering a holistic framework to comprehend and optimize the use of these convergent tools in programming.