First of all, you should notice that the “Lost Coaster” isn’t one coaster, but a twin pair of coasters. They should cover the VIP goal for apprentice, just ensure Major Smythe rides both coasters. Once you set the VIP’s path, deal with the freely roaming (or rather crawling) black panthers. Sedate them and build a medium sized chain fence enclosure somewhere not too far from the entrance and build some viewing galleries for some profit. Also, build at least two carnivore houses and place the panthers you just caught in the enclosure. Their health is very low so there is a risk the Animal Protection Society will confiscate some of them, don’t mind it too much, you can buy more panthers if necessary. Ensure they have enough food and water (in the carnivore houses) and their health should start going up slowly.
While the panthers are getting better, start building flat rides and shops in the area near the entrance. You could also build another coaster, just ensure it has at least excitement rating 5. When you have the money for it, you could build the other required coaster, or more if you want. I built a custom Junior Roller Coaster, a Bobsleigh and an additional Corkscrew Coaster but any coasters are fine as long as they have the 5+ excitement rate. Just note that Major Smythe doesn’t like to walk too much. Therefore, when you set his path to visit the coasters, try to keep that in mind. This way, he shouldn’t go back and forth, because else, he might will not ride all of the coasters. With some flat rides and the coasters, the park value goals should be accomplished too.
At some point you will likely get some notices that P. the panther got pregnant. The panthers' gestation period is only 3 months and the average litter is 3, so very soon, you’ll get a lot of baby panthers. However, their maturity age is 29 months, and you can’t release baby animals into the wild. So, if you plan to fulfill the panther objective with those babies, you have to wait a lot. Meanwhile the panthers will breed like rabbits, so soon the Park Inspector will start pestering you about P. the panther being unhappy because there are too many animals in its enclosure. This is an important info, as their health will drop to about 90% and won’t rise. And you can’t release them into the wild if their health is under 95%. You could sell some of the panthers, but a better idea could be building a separate chain fence enclosure with carnivore house. Put 2 or 3 of your panthers there that you plan to release into the wild. Ensure their health is good and release them. Do this as many times as necessary. A good idea saving time would be to buy new panthers instead of waiting for the babies to become adults. Just put the new panther into the second enclosure and ensure it has food, water, as well as some company (one of your panther you don’t plan to release at least yet). That way its social requirement is met, and, as soon as its health is more than 95%, release it to the wild. Do it with as much panthers as necessary and you won apprentice, entrepreneur and tycoon (assuming you fulfilled the VIP’s coaster visit objectives).