Let **a, b, c, d** and **e** are real positive numbers, such that

{a, b, c, d, e} > 0 and **\(a\neq b\neq c\neq d\neq e\)**.

Now consider following properties of {a, b, c, d, e}

1. **a, b, c** in A.P. (Arithmetic Progression)

2. **b, c, d** in G.P. (Geometric Progression)

3. **c, d, e** in A.P. (Arithmetic Progression)

4. **\(a^{2}\)** **+ e = [ a \(\times\) b ] + d**

5. **c + d + e = [ 2 \(\times\)** **\(a^{2}\) ]**

If **[ a + b + c + d + e ]** is represented in terms of **\(\frac{p}{q}\)**, where **p** and **q** are co-primes.

What is value of **[ 3 \(\times\) q ] - p**.

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