This puzzle replaces all numbers with different symbols.

Your job, because the title suggests, is to seek out what worth suits within the place of $bigstar$. To get the essential concept, I like to recommend you remedy Puzzle 1 first.

All symbols comply with these guidelines:

1. Each numerical image represents integers and solely integers. This means fractions and irrational numbers like $sqrt2$ aren’t allowed. However, unfavorable numbers and 0 are allowed.
2. Each image represents a distinctive quantity. This implies that for any two symbols $alpha$ and $beta$ within the puzzle, $alphaneqbeta$.
3. The following equations are glad (that is the center of the puzzle):
   $$
   textual content{I. }atimes a=a

   area

   textual content{II. }c<b<a

   area

   textual content{III. }b^c<c^b

   area

   textual content{IV. }dtimes b+b=c

   area

   textual content{V. }dtimes d+c+c<btimes b<d+d

   area

   textual content{VI. }e+btimes btimes e=d

   area

   textual content{VII. }a+btimes b+c+d+d+e=bigstar
   $$

What is a Solution?

An answer is a worth for $bigstar$, such that, for the group of symbols within the puzzle $S_1$ there exists a one-to-one operate $f:S_1toBbb Z$ which, after changing all offered symbols utilizing these capabilities, satisfies all given equations.

Can you show that there’s just one potential worth for $bigstar$, and discover that worth?

Side Note: to get $bigstar$ use $bigstar$, and to get $textual content^$ use $textual content^$

Previous puzzles:

Introduction: #1 #2 #3 #4 #5 #6 #7

Inequalities: #8 #9

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