Coin Exchange

Lorenzo of Valtier is a traveling merchant navigating the fractured realms of the Five Kingdoms. Each nation mints its own unique coins using secretive forging techniques, creating a hidden economy of conversions:

Currencies

• Aetherspire Dominion: Obsidian discs etched with celestial runes (Aetherspire Coin)
• Bleakmarch Protectorate: Bone-white tetrahedrons that scream when heated (Bleakmarch Coin)
• Crimson Falconate: Blood-veined square that warm before storms (Crimson Coin)
• Drowned King’s Reach: Barnacle-encrusted hexagons that smell of brine (Drowned Coin)
• Emberveil Syndicate: Geometric amber prisms containing frozen flames (Emberveil Coin)

Exchange Rates (Black Market, One-way Currency Exchange):

• $3$ Aetherspire $→$ $1$ Bleakmarch ("Three stars bow to the Pale Lord")
• $3$ Bleakmarch $→$ $1$ Crimson ("The Bloodied Falcon’s Toll")
• $5$ Aetherspire $→$ $1$ Crimson ("Stardust to Blood" smuggling route)
• $3$ Emberveil $→$ $2$ Drowned ("Fire drowns in black waters")
• $3$ Bleakmarch $→$ $4$ Emberveil ("Bleaching the Pale Mark")

Forbidden Technique: The Argentum Revenant Tome allows you to exchange $2$ Drowned $→$ $3$ Bleakmarch ("Raising Drowned Silver") for a maximum of $X$ uses, after which the Pale Inquisition is triggered.

Given Lorenzo of Valtier’s initial coin stash and a limit of $X$ on the number of times the forbidden technique may be used, your goal is to maximize the number of Crimson Falconate coins that can be obtained using a series of exchanges. Solving this puzzle may unlock the secret behind how Lorenzo of Valtier became wealthy!

Figure 1: Echange Rates. (Coin images created by DALL·E-3)

A line containing six integers: $A$ (Aetherspire Coin), $B$ (Bleakmarch Coin), $C$ (Crimson Coin), $D$ (Drowned Coin), $E$ (Emberveil Coin), $X$ (maximum use of the Forbidden Technique) satisfying $0≤A,B,C,D,E,X≤10^9$.

A single integer indicating the number of maximum possible Crimson Coins that can be obtained using a series of zero or more of exchanges.