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Only three words put K first and Y third. There is no branch to work through, only three entries.
KAYAK is the one most people would produce, and it reads the same forwards and backwards. KEYED is the other ordinary word. KAYOS, a plural of the knockout sense of kayo, is the third.
A takes the second box in two of the three and E takes the other. The Y then sits third in every case, doing the job a vowel would normally do.
Each of the three ends on a different letter and each takes a different fourth letter, so any single confirmed box settles the page outright.
Nothing here repeats an adjacent letter, so a visible double rules the pattern out entirely even though KAYAK uses its K twice.
Yes, it reads the same in both directions. That makes it easy to remember and easy to check once you have any two boxes confirmed.
A plural of kayo, meaning a knockout. It is the least likely of the three and the only one ending in S.
K is a rare opening letter and a Y in the third box is rare after anything. The two constraints together leave almost nothing.