After installing Security patch 10415, both frontend and backend return 404 error. How can I solve this error?
Note: No log error found.
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You need to make sure you have all previous security patches applied or there will be conflicts like the ones listed above.
hunk FAILED error is when the content within the file being patched doesn't match up with what the patch is trying to replace.
I would test your site on http://magereport.com/ and see which patches it flags up and then install them.
If you are sure about installing all the previous patches of this patch(PATCH_SUPEE-10415_CE_126.96.36.199_v1-2017-11-27-06-50-52.sh) try to compare the code diff in the patch file to that of the file where the hunk is failing.
In your case it is failing at app/code/core/Mage/Adminhtml/Controller/Action.php. Open that file and compare changes with the patch file, apply it manually and remove that diff from the patch file and again run the patch. Repeat it till the fails are zero. It will help sometimes.