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She let out a sigh and clenched both hands tightly.
Literal
She [topic-は] sigh [object-を] put-out [and-て] both-hands [object-を] firmly clenched-tight.
Two stock collocations and a compound verb pack a lot of physical detail into one sentence. ため息をつく ('let out a sigh') uses つく — the same versatile verb for 'attach, put forth, emit' that gives 嘘をつく ('tell a lie') and 悪態をつく ('curse'). 握り締める is a 複合動詞 (compound verb) built from 握る ('grip') + 締める ('tighten, fasten'), creating a compact word for 'clench tightly.' しっかり is a state mimetic (擬態語) for firmness or solidity — held tight, gripped securely, fastened thoroughly. The te-form ~て chains the sigh and the clenching as sequential gestures. The sentence reads like a stage direction: a single beat of physical reaction.