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Make sure you don't just follow this blindly. It can be flexible. Mana stone and two potions is a great starting point to give you more mana to harass with and potions to heal between altercations. Remember that these potions suck and will most likely not keep you alive in a fight. They are to replace health between fights.
Catalyst is obviously to keep you in fights longer and the boots are standard mage boots designed to help get you kills early on.
I prefer to get rod of ages first so it will max out its bonus around mid game. Pick is next to generate money flow.
Scepter and Ring provide good damage and great survivability. Void Staff makes your bombs hit SOOO hard and can be bought in between Scepter and Ring if you feel your survivability is good. If your really getting a lot of kills Void Staff after your shoes to really help your bombs as well and can continue your streak.
Deathfire grasp is a personal favorite because of its active nuke. Late game your bombs just won't be doing as much damage because of the team fights. Infact its sometimes better to bomb one champion -> rewind -> bomb a different champion if your team is staying alive long enough for both bombs to go off. Death fire grasp provides you with another nuke while your bomb is on cooldown. If the team fights are fast paced then definitlty go for burst damage to get kills and give your team advantages by doing the bomb -> rewind -> bomb combo on a single champion.
Flash is a must have skill. Creating space while your cooldowns reset is what wins Zil and your team fights. Additionally it can be used to catch running foes to apply your slow. Or to flash in and place a bomb as your enemy reaches a tower only to find you speed up and run away.
Teleport is a personal favorite right now because it can be used dynamically. It gets you back into your lanes faster after buying items or should you die. It also allows you to pick up slack and change lanes quickly. Is great for setting up ganks by teleporting to totems minions that are behind pushing enemies or located in bushes etc.
Ignite is also a very good skill especially against squishy enemy champions. A personal favorite is to Bomb -> Ignite -> Time Warp on Zil -> Rewind -> Bomb. The burst damage from the first bomb is usually very significant and ignite softens the enemy up enough to allow your second bomb to get the kill.
Exhaust is a good slow spell but since you do already have one it may not be necessary. If your having trouble getting kills mid game because people are getting away Time Warp -> Bomb -> Rewind -> Bomb -> Time Warp -> Bomb -> Exhaust -> Bomb will definitely do the trick. If you do choose exhaust obviously swap the one point in deadliness for improved exhaust.
Clarity sounds good in theory but I prefer the flexibility of healing/regening mana and getting back into the fight quick with teleport.
Magic penetration + Starting AP + leveling your bombs quick is a very scary combination. Often at level 3 or 5 you can easily get a first blood because enemies have a hard time comprehending how good Zils bomb combo can be.
If you get mid good for you. Harass with your bombs they have good range and keep in mind that you don't always have to rewind and bomb again. One bomb will often scare the enemy out of range and allow you to fully finish the cooldown. The Bomb -> Rewind -> Bomb combo is very good damage but you have to keep in mind that it costs a lot of mana.
If your in a lane with another allied champion try to get one with a stun or slow or pull.
Sion, Blitzcrank, Aniva, Rammus, Amumu are some of my favorites to lane with because stuns, slows, blocks almost guarantee you kills and help you escape while pulls like blitzy can snag champions when they are low and hugging turrets after you've been bomb harassing.
Help set up ganks and be cautious of other lanes mia's. Your more of a support class so help speed up your allied champions to catch stragglers. Definitly boost your speed to bomb if you can.
Think support and you will be ok. Continue with your bomb combo and keep an eye on allies that have low health for your ult and allies that are trying to catch or run away from enemies so you can slow / speed up accordingly.
What seperates a great Zil support player is the use of his ult. You have to make a quick judgement call sometimes on who to save. The following is my personal preference of priority targets for your ult.
1. Champions with good escape mechanisms and survivability and have oracle.
2. Champions with good escape mechs and surviv
3. Tanks with oracle.
5. Highest DPS ally.
7. All other classes.
Essentially what I am trying to achieve with a prioritization is a way to maximize the odds of a champion actually living through its second life and making it to safety or in team fights tanks healers and high dps allies to get your team the win in the fight.