These used to plague me to no end (before I started reading). But now when I see them, since they show up so often, I have a feel for the specific meaning of each one. As you say, it becomes clearer sooner or later.
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The first thing for this panel is I think the two thought balloons are separate sentences. (I don’t recall if anyone mentioned this already.) The first sentence is “unfinished” because it’s clear what would follow (at least to a Japanese reader?) Unfinished sentences perplexed me initially, until I realized we use them in English as well (although not nearly as frequently).
That said, what’s with 「(noun)しようと」? We have 「(noun)する」 meaning to do (the action of) the noun. And する is in the volitional form, しよう. This is followed by と, which often has a verb after such as 言う or 思う, alongside various other possibilities.
For 思う, I learned at one point that this can only be used for one’s own thoughts, but I’ve never paid attention when reading to see if it gets used to express someone else’s thoughts. Thus, I’d consider this likely isn’t 思う, with the understanding that I may be wrong.
Another possibility is する is the unspoken verb. I thought I saw this mentioned above, but don’t see it offhand. In this situation, と+する is used to mean “trying to do”. To expand on that:
Consider the sentence:
「ちゃんと手紙で返事しよう」
Here, the verb at the end is “intend to respond” (返事 + to do + volitional).
Next is the quote-marking と:
「ちゃんと手紙で返事しよう」と
And finally add our verb at the end:
「ちゃんと手紙で返事しよう」とする
This gives a meaning something like:
To do “intend to properly respond by letter”.
When saying someone is doing an intended action, the meaning is that they are attempting to do something.
「しかも ちゃんと手紙で返事しようと」
“What’s more, he’s attempting to properly respond by letter.”
Note that this is the same meaning as with 思う, except there instead of “attempt to do”, it’s “think to attempt to do”. For example, if you try something and it fails, you might state what you intended and add と思った to say you “thought you’d attempt” it.
So, is する the missing verb here? I’m not certain, but that’s what I’m leaning toward. If it were Ayumu himself speaking, I’d learn a bit more toward 思う.
I believe the “posted deleted by” only shows for 24 hours, then the post removes completely. No need to try any clean-up on what’s already been posted (unless you want to), but now you know for future reference =D