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If you are wanting the Amazon store card because you will use it, go for it. I have the Amazon Prime Visa Signature card from Chase. Maybe check that one out instead of the store version from Sync. I use it just for Amazon as I'm Prime and do a lot of purchases on Amazon but being it's the Visa version, I can use it anywhere. I also got it when they had the free $70 gift card too! Not sure what promotion the Chase version may have currently if any though. Your recent Amex approval is great! If you do decide to snag the Amazon store card, or maybe the Visa version, I'd definitely hit the garden and let what you have grow!
If you really want the Amazon, get it and garden for a while. Like one poster mentioned, if you want the Chase version, its a Visa and can be used anywhere. Where are your scores from, myfico, creditkarma? Also, if you are planning on getting a mortgage, chill for a year.
You can ask for a CLI on the Amazon store card via chat. They're very generous with giving CLIs