To describe a DOK-4, extended thinking task, the most appropriate words from your list would be:
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assess - This implies evaluating or judging the value or significance of something, which aligns with the deeper analysis and critical thinking characteristic of DOK-4 tasks.
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prove - This suggests providing evidence to demonstrate the truth of something, which involves argumentation and justification, fitting well within the extended thinking nature of DOK-4.
The other options, list and compare, are more aligned with DOK-1 and DOK-2 tasks, and do not adequately capture the complexity and depth required for DOK-4 tasks. DOK-4 requires tasks that encourage students to engage in long-term, self-designed projects that require synthesizing knowledge, making connections, and applying reasoning over a period of time.