Explanation:

This question asks to identify the correctly spelled word in Hindi (शुद्ध शब्द). This tests knowledge of Hindi orthography (vartani). Let's analyze each option: Option (1): "संन्यास (Sannyas)". This word means 'renunciation' or 'asceticism'. The correct spelling in Hindi is 'संन्यास', which involves a 'bindu' (अनुस्वार) followed by a half 'न' (न्) and then 'यास'. The option 'संन्यास' is **correctly spelled**. This is a common word tested for its precise spelling. Option (2): "कैलाश (Kailash)". This word refers to Mount Kailash, a sacred peak. The correct spelling in Hindi is 'कैलाश'. The option 'कैलाश' is also **correctly spelled**. This presents an ambiguity as there are two correctly spelled words among the options. Option (3): "सौंदर्यता (Saundaryata)". This word is incorrectly formed. 'सौंदर्य' (Saundarya) itself is an abstract noun meaning 'beauty'. 'सुंदरता' (Sundarta) is another correct abstract noun meaning 'beauty' (सुंदर + ता). Adding the suffix 'ता' to 'सौंदर्य' (which already means beauty) creates a redundant and grammatically incorrect form. Therefore, 'सौंदर्यता' is **incorrectly spelled/formed**. Option (4): "पूज्यनीय (Pujyaniya)". This word is incorrectly formed. The correct words are 'पूजनीय' (Pujaniya), meaning 'respectable' or 'venerable', or 'पूज्य' (Pujya), also meaning 'respectable' or 'worshipful'. 'पूज्यनीय' is a hybrid and grammatically **incorrect** form, combining parts of both 'पूज्य' and 'पूजनीय'. **Conclusion:** Both Option (1) 'संन्यास' and Option (2) 'कैलाश' are correctly spelled words. Options (3) and (4) are definitively incorrect. Since the question asks for *the* correctly spelled word and provides multiple choices, and two of them are correct, the question is ambiguous. However, in a multiple-choice format, a single answer must be provided. Often, one correct option might be considered more 'correct' or a more common test item. For the purpose of providing a single answer as required by the format, and acknowledging that both (1) and (2) are correct, I will select (0) as the 'correct_option_index' for the JSON, and explain that both (1) and (2) are correct while (3) and (4) are incorrect.