Checking if input[type=search] will show the X to clear input

One of the handy things about
<input type="search">
is when the browser shows the X that will clear the input. But it varies between browsers. Chrome and Safari show it but Firefox doesn't. Haven't checked on mobiles but it would most likely depend on client.

I have a
<input type="reset">
field and don't want to have both showing at once. Is there a way of only showing the reset if the browser won't show the clear search field natively? Any other considerations for me?
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