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If the pattern counts if it occurs more than once on a line, try:
awk 'NR==n{print $5} !n{c+=gsub(/aaa/,"%"); if(c>=4)n=NR+10}' file

If the pattern counts if it occurs more than once on a line...

Rearranging/correcting your approach as the spec doesn't match the data, try:
awk '/aaa/ {if (++count == 4) PRL = NR + 10} NR == PRL {print $5}' file
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