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I'm not sure if this logic is good but here is my assumption about Tsunade's decision making:
Tsunade assumes that akatsuki thinks Naruto has been taken to an unknown place to be protected and hidden, and that akatsuki will search the whole fire country for Naruto, especially the hidden fortress-like places. And that monk informed Tsunade that akatsuki began the search for Naruto, and they will probably destroy all these special places in search for the last demon. So it's logical that she wants to take them out before they find out where is Naruto and also before they destroy a lot of the fire country's special places. She can assume that akatsuki don't know of Naruto's current position while she knows of akatsuki's recent position, and that's a small advantage now on her side. It's also logical to send out a whole army, after she got the report that Deidara almost destroyed the whole sand village by himself. So that akatsuki must be stopped before they reach a place full of innocent people, a battle in Konoha would be a catastrophy.
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