Counterspeech effectively neutralizes the impact of online hate. Although prior work explores automated counterspeech generation, it largely emphasizes stylistic control while treating hate speech as homogeneous, overlooking that distinct forms of abuse require fundamentally different counterspeech strategies. To address this gap, we introduce FIRE (Factuality Informed Multi-Agent Reasoning Framework) that first decomposes hate speech into one of the five distinct categories (misinformation, stereotype, conspiracy, dehumanizing, non-factual), and then maps it to a targeted counterspeech style. To facilitate FIRE, we curate FactualCS, a novel dataset of $4,784$ instances that provides the annotations regarding hate categories, reasoning traces, and evidence mappings, which are critical elements for grounded generation that are missing in prior work. A comprehensive evaluation across $28$ baseline configurations demonstrates that FIRE significantly surpasses existing methods, despite using compact agents ($<$2B). FIRE achieves a $\sim$ $12 \%$ and $\sim$ $11 \%$ improvements in factual and category-specific accuracy respectively, while simultaneously reducing toxicity by $\sim$ $11 \%$ relative to the strongest baselines. Further human evaluation confirms that responses generated by FIRE are significantly preferred over the strongest baselines, underscoring its effectiveness for real-world deployment. These findings show that decomposing the underlying intent of hate speech is essential for generating safe, effective, and contextually precise counterspeech.

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