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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

Because the last ability modifies the rules of the game, it affects not only planeswalkers you control when it resolves, but also planeswalkers that come under your control later in the turn.

For the first ability, it doesn’t matter whether the planeswalker is still on the battlefield as your end step begins. If you activated one of its loyalty abilities that turn, The Chain Veil’s triggered ability won’t trigger.

Each additional time The Chain Veil’s last ability resolves will allow you to activate a loyalty ability of each planeswalker you control an additional time. For example, if you activate The Chain Veil’s last ability, untap it, then activate it again, you can activate a loyalty ability of a planeswalker you control three times that turn.

After the last ability resolves, you’ll essentially be able to activate a loyalty ability of each planeswalker you control a total of twice during your turn. The timing rules for when you can activate loyalty abilities apply each time; it must be your main phase and the stack must be empty.

The second loyalty ability you activate doesn’t have to be the same as the first ability. For example, you could activate a planeswalker’s first ability twice, or you could activate a planeswalker’s first ability, then activate its second ability.