a decomposition trajectory that leads to the correct answer. We establish a maximum limit of 10 rounds per question. The results of our data collection process prove to be efficient - we successfully obtain correct answers along with their corresponding decomposition trajectories for 84% of the training data, which represents a substantial improvement compared to the baseline accuracy of 58% achieved without implementing context sampling techniques. \n\nTraining Method Our training methodology is designed to optimize model performance by fo-cusing exclusively on decomposition trajectories that successfully lead to correct answers. The data structure for each record follows a format that captures the complete problem-solving path: \n\n(q, [( q1, a 1), . . . , (qt, a t)] , a ), where q and a are the original question and its answer, respectively. Each pair (qi, a i) in the trajectory is composed of an atomic question and its answer. This structured approach ensures that we maintain a clear record of the decomposition process from start to finish. For our LLM training, we employ supervised fine-tuning (SFT) to enhance model performance. Our process converts each trajectory into structured training data pairs of prompts and responses. These pairs are formatted as (xi, y i) according to Algorithm 3. The algorithm incorporates two key functions: prompt x and prompt y , which generate contextual descriptions based on input parameters. These functions are integral to the transformation process, and we detail their implementation in the following sections. 30 Algorithm 3 Transform each decomposition trajectory into data pairs for SFT \n\n> 1:\n\nLet (q, [( q1, a 1), . . . , (qt, a t)] , a ) be the given decomposition trajectory. \n\n> 2:\n\nInitialize an empty list to store pairs of prompts and responses Dsft = ∅. \n\n> 3:\n\n# Prepare training data when a further decomposition is necessary. \n\n> 4:\n\nfor i = 1 , 2, . . . , t do \n\n> 5:\n\n# prompt x takes the original question plus any previously collected sub-questions and answers as input. \n\n> 6:\n\nxi = prompt x(q, [( q1, a 1), . . . , (qi−1, a i−1)]) \n\n> 7:\n\n# prompt y takes two inputs: a decomposition indicator and an optional sub-question. \n\n> 8:\n\nyi = prompt y (T rue, a i) \n\n> 9:\n\nDsft ←− (xi, y i) \n\n> 10:\n\nend for \n\n> 11:\n\n# Prepare training data when no further decomposition is necessary. \n\n> 12:\n\nxt+1 = prompt x(q, [( q1, a 1), . . . , (qt, a t)]) \n\n> 13:\n\nyt+1 = prompt y (F alse, N one ) \n\n> 14:\n\nDsft ←− (xt+1 , y t+1 )\n\nPrompt Template for prompt x\n\n> Based on the given information, determine whether a follow-up question is necessary or not. **Original Question** {the original problem} **Existing Context** {A list of sub-questions and their answers} Make sure your output align with the following format: False OR True a follow-up question\n\nPrompt Template for prompt y if a sub-question is provided \n\n> True {The given sub-question}\n\nPrompt Template for prompt y if no sub-question is provided \n\n> False\n\nThis transformation is particularly efficient because it generates multiple training instances from each trajectory - specifically, for a trajectory containing t sub-questions, we generate t + 1 distinct training data points, maximizing the learning potential from each successful decomposition. For all experiments, we use a learning rate of 1.5e−5 and parameter-efficient fine-tuning (PEFT) with LoRA configuration parameters (lora=16, alpha=64). To ensure reproducibility, we maintain all other hyperparameters at their default values as specified in reference [ 56 ]. We run all experiments three times with different random seeds and report the mean results. We perform the experiments on a computing node equipped with a single NVIDIA A100-80G GPU for all training processes. \n\nExperiment Results Table 11 presents our experimental findings. We select

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