"To lay" has many meanings including to put down with care as in "She laid the baby in the crib." For this English quiz, "to put" is natural and "to lay" is awkward. In a different context, lay would be natural. "He was reading a book of poetry when a stranger walked up. He laid the book on the table, and then he looked up at the stranger." New Oxford American Dictionary lay1 | leɪ | verb (past and past participle laid | lād | ) 1 [with object and adverbial of place] put down, especially gently or carefully: she laid the baby in his crib. • prevent (something) from rising off the ground: there may have been the odd light shower just to lay the dust. 2 [with object] put down and set in position for use: it is advisable to have your carpet laid by a professional | figurative : the groundwork for change had been laid. • British set cutlery, crockery, and mats on (a table) in preparation for a meal: she laid the table for the evening meal. • (often be laid with) cover (a surface) with objects or a substance: the floor was laid with tiles. • put the material for (a fire) in place and arrange it. • make ready (a trap) for someone: she wouldn't put it past him to lay a trap for her. • work out (an idea or suggestion) in detail ready for use or presentation: I'd like more time to lay my plans. 3 [with object] used with an abstract noun so that the phrase formed has the same meaning as the verb related to the noun used, e.g., “lay the blame on” means ‘to blame’: she laid great stress on little courtesies. 4 [with object] (of a female bird, insect, reptile, or amphibian) produce (an egg) from inside the body: flamingos lay only one egg | [no object] : the hens were laying at the same rate as usual. 5 [with object] vulgar slang have sexual intercourse with. 6 Nautical follow (a specified course).

Lying "lying down" pictures of people and animals "lying down" https://pixabay.com/en/photos/lying%20down/ LAYing "LAY tiles" picture "of tile being laid" https://pixabay.com/en/photos/laying%20tiles/ man putting book on a shelf https://www.alamy.com/stock-photo-worker-putting-box-on-shelf-71597581.html picture men "putting" price labels on cartons http://www.wmky.org/post/kentucky-releases-unemployment-data-april-2018 video "putting books" on a table https://us.clipdealer.com/video/media/848205 "put down" in your context means. The book is so good you will not want to stop reading it. "put it down anywhere"

No it is not possible to use lay down in that context. Lie and lay are two words that confuse learners, They are frequently used incorrectly by native speakers. Lie= 1) to tell an untruth. Lie =2) for a person or animal to lie down somewhere on a bed on the grass etc. Lay. = to place objects onto a surface that makes a covering over or onto that surface. you lay tiles and carpets on floors or tiles and slates on roofs. or grass turfs on a lawn.etc used in building construction generally lay concrete to make a path or a floor to lay wooden floors onto concrete etc. For all other things you "put" onto a surface or into a space, cupboard etc. Put : book onto a table. "food in a cupboard" "milk in a fridge" "shoes on the floor" .