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I'm not sure how well this meets all your somewhat conflicting criteria, but take a look at Mt Taylor northeast of Grants. Grants is right on the interstate, and its not that far into the national...
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#1: Attribution notice removed
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System
·
2020-06-03T15:54:21Z (over 4 years ago)
Source: https://outdoors.stackexchange.com/a/10762 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/