Meaning of "4歳と14歳で、生きようと思った"


I'm trying to breakdown the tagline for the movie [火垂るの墓 in a movie
poster](https://pbs.twimg.com/media/CrUUt2GVUAAs0W5.jpg): "4歳と14歳で、生きようと思った"

> 4歳と14歳で

My first guess was that the で here is the て-form of the copula だ, so it just
joins that clause to the next.

But the whole sentence makes more sense if it's the particle で that expresses
a cause, as in, "since they were (only) 4 and 14 years old, ...", but it seems
like a strange place to put a comma.

> 生きようと思った

The structure [volitional + と思う] here, as I understand from the [_Dictionary
of Basic Japanese Grammar_ (p.569)](https://i.stack.imgur.com/fcb2Q.png),
indicates the speaker's desire, or what the speaker intends to do.

But I'm not sure how to map that explanation to this sentence.

My best guess here is that the tagline means, "Since they were (only) 4 and
14, I thought that they would live", but I feel like I've misunderstood this
sentence.

There are a few other threads about ~おうと思う, but I don't think they answered my
question.

Thanks for the help.




で can mark a cause/reason (eg 戦争 **で** 多くの人が死んだ), but "to think we will
survive because we are 4 and 14" doesn't make sense to me.

I feel this で is the same で as in 1人で生きる ("to live alone"), 全員で考える ("to think
together"), 個人で検討する ("to investigate individually"), etc. で marks the (number
of) people involved in the action. Broadly speaking, it's referred to as an
amount marker [here](https://www.wasabi-jpn.com/japanese-grammar/particle-de-
expressing-supplementary-information/#6).

Note that ～歳で usually means "at the age of ～" (eg 25歳で結婚する "to marry at the
age of 25"). However 4歳と14歳で生きよう refuses this interpretation, and people would
imagine something like 14歳(の少年)と4歳(の少女)の2人で生きよう when seeing it, and understand
how isolated they are. The catchphrase has been abbreviated aggressively
because it's a catchphrase, so you don't have to think it is a common regular
usage of 歳で.

