Lanette's Net Search can find a dual type and then 2 other pokémon (searching for 4 or more types)

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As said above, Lanette’s can search for Jirachi (Psychic/Metal), Larvitar (Fire) and Electrode (Lightning), breaking the following rules:

  1. Q. Lanette’s Net Search says to search for 3 different types of Pokémon. If I get a Fire-Darkness Pokémon, does that count as 2 of those cards? A. Yes. If you chose a Dual type Pokémon, you could only choose 1 more Pokémon (and it couldn’t be either of those two types). (Dec 16, 2004 PUI Announcements)

  2. Q. If I use Lanette’s Net Search to search for a Dual Type Pokémon (like Team Magma’s Groudon, Darkness/Fighting), can the second Pokémon searched for also be a dual type (such as Jirachi, Metal/Psychic)? A. No, the total number of TYPES you can search for is three and those types cannot be duplicated. So if you tried searching for two Dual Type Pokémon, that would be four types which is one too many. (Dec 16, 2004 PUI Rules Team)

https://compendium.pokegym.net/compendium-ex.html

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Umbreon is correct. The Japanese ruling archived here agrees with the English Compendium.