In automated web testing, generating test scripts from natural language task descriptions is crucial for enhancing the test generation process. This activity involves creating the correct sequences of actions to form test scripts for future testing activities. Current state-of-the-art approaches are limited in generating these action sequences, as they either demand substantial manual effort for human demonstrations or fail to consider the history of previous web content and actions to decide the next action. In this paper, we introduce HxAgent, an iterative large language model agent planning approach that determines the next action based on: 1) observations of the current contents and feasible actions, 2) short-term memory of previous web states and actions, and 3) long-term experience with (in)correct action sequences. The agent generates a sequence of actions to perform a given task, which is effectively an automated test case to verify the task. We conducted an extensive empirical evaluation of HxAgent using two datasets. On the MiniWoB++ dataset, our approach achieves 97% exact-match accuracy that is comparable to the best baselines while eliminating the need for human demonstrations required by those methods. For complex tasks requiring navigation through multiple actions and screens, HxAgent achieves an average 82% exact-match. On the second dataset, comprising 350 task instances across seven popular websites, including YouTube, LinkedIn, Facebook, and Google, HxAgent achieves high performance, with 87% of the action sequences exactly matching the ground truth and a prefix-match of 93%, outperforming the baseline by 59%.