League of Legends (LoL) is a multiplayer online battle arena game developed by Riot Games. Set in the fictional world of Runeterra, players assume the roles of "champions," each with unique abilities and roles within teams of five. The objective is to work together to destroy the opposing team's Nexus, a heavily guarded structure within their base while defending their own.

Champion Overview: Katarina
Katarina, the Sinister Blade, is an assassin known for her high burst damage and mobility. She excels at quickly eliminating priority targets in team fights and skirmishes, relying on precise timing and positioning to maximize her effectiveness. 

Champion Details: Annie and Zed
Annie
Annie, the Dark Child, is a burst mage who combines high burst damage with crowd control. She is relatively straightforward to play but requires good positioning and timing to unleash devastating combos. When you embark on your journey, mmowow items can lend a helping hand when you need help.

Abilities:

Passive - Pyromania: Annie's offensive abilities stun enemies after every four spell casts.
Q - Disintegrate: Annie casts a fireball at a target, dealing magic damage and refunding mana if it kills the target.
W - Incinerate: Annie releases a cone of fire in front of her, dealing magic damage to all enemies hit.
E - Molten Shield: Annie shields herself from damage and increases her movement speed, while also dealing magic damage to enemies who attack her.
R - Summon: Tibbers: Annie summons Tibbers, her bear, dealing magic damage to nearby enemies and causing Tibbers to attack enemies.
Skill Order:

Max Q first for poke and last hitting.
Max W second for AoE damage.
Max E lasts for survivability and utility.
Rune Recommendations:

Primary (Domination): Electrocute, Cheap Shot, Eyeball Collection, Ultimate Hunter.
Secondary (Inspiration): Magical Footwear, Cosmic Insight.
Stat Shards: Adaptive Force, Adaptive Force, Magic Resist/Armor depending on the matchup.
Item Build:

Core Items: Hextech Rocketbelt, Sorcerer's Shoes, Zhonya's Hourglass.
Situational Items: Rabadon's Deathcap, Void Staff, Banshee's Veil.
Boots: Sorcerer's Shoes.

Gameplay Strategy:

Early Game: Focus on farming with Q and harassing with Q-W combo when possible. Use E to mitigate damage and set up ganks.
Mid Game: Roam to other lanes and use R in team fights for AoE damage and crowd control. Look for picks with Q and W.
Late Game: Position yourself carefully in team fights. Use Zhonya's Hourglass after engaging with R to survive and continue dealing damage.
Zed
Zed, the Master of Shadows, is an AD assassin known for his high single-target burst damage and outplay potential. He relies on energy instead of mana and excels at quickly assassinating high-priority targets.

Abilities:

Passive - Contempt for the Weak: Zed's basic attacks against targets below 50% health deal bonus magic damage.
Q - Razor Shuriken: Zed throws a shuriken forward, dealing physical damage to enemies in its path.
W - Living Shadow: Zed’s shadow dashes forward, mimicking his basic abilities. Reactivating W sends Zed to his shadow's location.
E - Shadow Slash: Zed slashes in a nearby area, dealing physical damage to nearby enemies.
R - Death Mark: Zed becomes untargetable and dashes to a target champion, marking them. After a delay, the mark triggers, dealing physical damage based on the damage dealt while marked.
Skill Order:

Max Q first for poke and wave clear.
Max E second for additional damage and slow.
Max W last for mobility and utility.

Rune Recommendations:

Primary (Domination): Electrocute, Sudden Impact, Eyeball Collection, Ultimate Hunter.
Secondary (Precision): Triumph, Coup de Grace.
Stat Shards: Attack Damage, Adaptive Force, Magic Resist/Armor depending on the matchup.
Item Build:

Core Items: Eclipse, Ionian Boots of Lucidity, Ghostblade.
Situational Items: Edge of Night, Guardian Angel, Lord Dominik's Regards.
Boots: Ionian Boots of Lucidity.

Gameplay Strategy:

Early Game: Farm safely with Q and use W for harass and escape. Look for opportunities to all in with the W-E-Q combo.
Mid Game: Roam to other lanes and look for picks with R. Use shadows to confuse opponents and escape unfavorable situations.
Late Game: Split-push and flank in team fights. Use R on priority targets and use W to reposition or escape. Utilize items like Edge of Night and Guardian Angel to survive burst damage.
Recommended Team Compositions
Annie
Aggressive Composition: Pair with engaged champions like Amumu (jungle), Malphite (top), and Jhin (ADC) to capitalize on crowd control.
Poke Composition: Works well with champions like Jayce (top), Ezreal (ADC), and Lux (support) to poke down enemies before engaging.
Team Fight Composition: Fits with champions who excel in team fights, like Sejuani (jungle), Orianna (mid), and Miss Fortune (ADC).
Strengths: High burst damage, AoE stun, relatively easy to play.
Weaknesses: Low mobility, relies on flash for engagement, susceptible to crowd control.

Zed
Assassin Composition: Pair with champions like Kha'Zix (jungle), Fiora (top), and Kai'Sa (ADC) to quickly eliminate priority targets.
Split Push Composition: Works well with champions who can pressure side lanes, like Camille (top), Nidalee (jungle), and Vayne (ADC).
Pick Composition: Fits with champions who can catch enemies out of position, like Thresh (support), Twisted Fate (mid), and Ashe (ADC).
Strengths: High mobility, strong single-target burst damage, outplay potential.
Weaknesses: Energy management can be tricky, and susceptible to crowd control and burst damage.

Laning Against Vayne
Annie vs. Vayne
Early Game:

Laning Phase: Use Q to poke Vayne from a safe distance and farm minions. Look for opportunities to stun with Q and R for an all-in combo.
Trades: Engage with Q-W combo when Vayne mispositions. Use E to mitigate return damage and set up ganks.
Mid Game:

Roaming: Use R to roam and apply pressure to other lanes. Look for picks with a Flash-R combo and follow up with Q-W.
Late Game:

Team Fights: Position yourself to land a multi-man R and stun with Q. Use Zhonya's Hourglass to survive after engaging.
Zed vs. Vayne
Early Game:

Laning Phase: Farm safely with Q and harass with W-E-Q combo when Vayne goes for CS. Avoid extended trades due to Vayne’s kiting potential.
Mid Game:

Roaming: Roam to other lanes and use shadows to engage with R. Look for picks and objectives.
Late Game:

Split Pushing: Use W to escape and R to engage on priority targets. Utilize items like Edge of Night and Guardian Angel to survive burst damage.

Detailed Analysis of Laning Against Katarina: Annie and Zed
Laning against Katarina presents unique challenges and opportunities for both Annie and Zed throughout the different stages of the game in League of Legends. However, when some players explore further and become familiar with a certain hero, some players will choose to buy cheap lol items.

Annie
Early Game:

Laning Phase: Against Katarina, Annie should prioritize poking with Q - Disintegrate while maintaining safe positioning to avoid Katarina's Q - Bouncing Blades and E - Shunpo engage. Stacking Annie's passive, Pyromania, is crucial to threatening Katarina with stun combos.
Trades: Engaging with the Q-W combo when Katarina overextends can force her to retreat due to Annie's burst potential. Use E - E-E-Molten Shield to mitigate damage if Katarina attempts to trade back.
Mid Game:

Roaming: As Annie, roam to other lanes with R - Summon: Tibbers to impact skirmishes and secure objectives. Look for opportunities to catch out Katarina or other squishy targets with the Flash-Tibbers combo.
Team Fights: Positioning is key; use Flash-Tibbers to initiate team fights and lock down multiple enemies, preventing Katarina from freely maneuvering and using her resets.
Late Game:

Team Fights: Stick close to your team to avoid being isolated by Katarina's mobility. Use Tibbers to zone and peel for your carries, and save E for absorbing damage or setting up stuns on Katarina when she dives.
Zed
Early Game:

Laning Phase: Zed should focus on farming with Q - Q-Razor Shuriken while avoiding Katarina's harassment. Use W - Living Shadow to poke and create pressure, forcing Katarina to use E - Shunpo defensively.
Trades: When Katarina uses E - E-E-Shunpo, follow up with a W-E-Q combo to harass and potentially all-in. Save W for escapes if Katarina tries to retaliate.
Mid Game:

Roaming: Zed's strength lies in roaming with W and using R - Death Mark to secure kills on Katarina or other vulnerable targets. Coordinate with your team to capitalize on your pick potential.
Team Fights: Look for opportunities to flank and assassinate Katarina or other high-priority targets with R. Use shadows to reposition and escape after engaging.
Late Game:

Split Pushing: Utilize Zed's split-pushing potential with W for escapes and R to quickly eliminate isolated targets like Katarina. Build items like Edge of Night and Guardian Angel to survive and outplay in team fights.
Analysis Based on Champion Attributes and Playstyle
Annie's straightforward burst and crowd control make her a strong pick against Katarina:

Attributes: Annie's passive Pyromania allows for reliable stuns, crucial for controlling Katarina's mobility.
Playstyle: Focus on stacking Pyromania and using R for team fight initiation. Position carefully to maximize Tibbers' impact.
Zed's mobility and single-target burst make him a viable counter pick:

Attributes: Zed's W provides versatility and escape options against Katarina's engagement.
Playstyle: Play aggressively in the planning phase, utilizing R to eliminate Katarina and split push effectively.

Conclusion
In conclusion, both Annie and Zed offer unique strategies to counter Katarina throughout the game. Annie excels with her reliable crowd control and burst damage, making her a solid choice for team fight control. On the other hand, Zed's mobility and assassination potential allow for aggressive playmaking and split-pushing advantages. By mastering their respective abilities and adapting to Katarina's movements, players can effectively control the tempo of the game and secure victory on the Rift.