You cannot add new language strings to the custom language packs that do not already exist in the English language pack built into the Moodle code.
If you have a custom theme, then I would suggest that you override the mustache template in your theme, add the language strings to theme/mytheme/lang/en/theme_mytheme.php and, in the mustache template, add: {{#str}}forgottenst, theme_mytheme{{/str}} (this is assuming your custom theme is called 'mytheme').
If you are not using a custom theme for the overrides and, instead, you are making changes to the core code, then you will need to add 'forgottenst' and 'forgottenc' to lang/en/moodle.php
If you have a custom theme, then I would suggest that you override the mustache template in your theme, add the language strings to theme/mytheme/lang/en/theme_mytheme.php and, in the mustache template, add: {{#str}}forgottenst, theme_mytheme{{/str}} (this is assuming your custom theme is called 'mytheme').
If you are not using a custom theme for the overrides and, instead, you are making changes to the core code, then you will need to add 'forgottenst' and 'forgottenc' to lang/en/moodle.php