Now, we have the following relations :

Arithmetic progression :

Since $A_1$ and $A_2$ are arithmetic means between $a$ and $b$, we can say that $a$, $A_1$, $A_2$, and $b$ are in an arithmetic progression. This means there are three equal intervals between $a$ and $b$, which are represented by the common difference $d$.

To find the value of $d$, we can use the following equation :

$$ b - a = 3d $$

From this equation, we can find the value of $d$ :

$$ d = \frac{b - a}{3} $$

$$ A_1 = a + \frac{b - a}{3} = \frac{2a + b}{3} $$

$$ A_2 = \frac{a + 2b}{3} $$

$$ A_1 + A_2 = a + b $$

Geometric progression :

$$ a, G_1, G_2, G_3, b \text{ are in G.P. } $$

$$ r = \left(\frac{b}{a}\right)^{\frac{1}{4}} $$

$$ G_1 = \left(a^3b\right)^{\frac{1}{4}} $$

$$ G_2 = \left(a^2b^2\right)^{\frac{1}{4}} $$

$$ G_3 = \left(ab^3\right)^{\frac{1}{4}} $$

We have the expression :

$$ G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2 = a^3b + a^2b^2 + ab^3 + \left(a^3b\right)^{\frac{1}{2}}\cdot\left(ab^3\right)^{\frac{1}{2}} $$

Simplify the expression :

$$ a^3b + a^2b^2 + ab^3 + ab(a^2b^2) $$

Factor out $ab$:

$$ ab(a^2 + ab + b^2 + a^2b^2) $$

Combine the terms :

$$ ab(a^2 + 2ab + b^2) $$

Rewrite the expression using the sum of squares :

$$ ab(a + b)^2 $$

Now, recall that $A_1 + A_2 = a + b$. Substitute this into the expression :

$$ G_1 \cdot G_3 \cdot (A_1 + A_2)^2 $$