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 clear up 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 $sqrt2$ should not allowed. However, detrimental numbers and nil 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 inequalities are happy (that is the center of the puzzle):
   $$
   textual content{I. }a^{a}-a<atimes a

   area

   textual content{II. }btimes b-a^{a}occasions b<a

   area

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

   area

   textual content{IV. }dtimes d-ctimes d+ctimes c<ctimes d+a

   area

   textual content{V. }atimes e^atimes c>d^{e}

   area

   textual content{VI. }etimes c<atimes b-atimes d

   area

   textual content{VII. }e^{a}<bigstar<atimes b^{a}
   $$

What is a Solution?

An answer is a price for $bigstar$, such that, for the set of symbols within the puzzle $S_1$ there’s a subtitution $f:S_1toBbb Z$ that satisfies all given equations.

Can you show that there’s just one doable 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 #10