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Sometimes when you are in battle you may find monsters attacking your pet and making its 'vital power' level drop. Your pet's 'vital power' level (the red bar) is basically its HP. If you let this bar drops to zero then your pet will faint. You must then go to the Animal Trainer and select 'Unseal Pet' to revive it. Unsealing your pet will cost more per level so unsealing a really high level pet repeatedly can get expensive.
Your pet's 'Degree of Starvation' is shown by the yellow bar, if this reaches zero your pet won't be able to move and will stop gaining experience from your battles.
There are two items you need to feed your pet in order to keep it healthy; green herbs and quality stones.
Green herbs can be found outside any city in the herb patches. You must buy a knife from the general vendor and then select a herb to begin harvesting, this will take a long time unless you have some Tool Aides. You can apply a Tool Aide by clicking on it and dragging it over the tool you are using. Tool aides are found pretty commonly throughout monster drops in most temples.They can also be bought from players selling items in the city centre.
You will also need quality stones to keep your pet healthy. First you need to by a mining tool, which varies depending on your class, from the general vendor, then go to a nearby mine and mine for quality stones to feed to your pet.
If you you find yourself without any tool aides and don't want to spend the time gathering the resources for you pet; there is a few players around the town centre that sell pet items, but they are notoriously expensive. Try to but some tool aids if possible, otherwise try to put aside some time for resource gathering.