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Our results are organized around the three primary hypotheses and a set of exploratory analyses. First, we examine whether conversations with sycophantic agents affect people’s chances of discovering the true rule. Second, we analyze individuals’ confidence levels across conditions. Third, we test whether conversations with the default GPT increased confidence in beliefs. Additional pre-registered exploratory analyses are omitted due to space constraints. Hypotheses and analyses were pre-registered prior to data collection (AsPredicted.org/94vn2y.pdf).444We deviated from the pre-registration in two ways: (1) Instead of excluding incomplete cases entirely, we used an LLM-based extraction method to recover partial data where possible. This was done to mitigate differences in completion rates across conditions. As a result, sample sizes vary slightly across analyses of discovery rates and confidence ratings (see Footnote 2 & 3). (2) We used permutation tests instead of the pre-registered Chi-square tests for H1. This provides a more conservative test of by avoiding distributional assumptions that may be unreliable given the low discovery rates.，这一点在下载安装汽水音乐中也有详细论述
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