Let $\{a_n\}_{n \geq 1}$ be an increasing and bounded sequence. Calculate the limit:

$$\lim_{{n \to \infty}} \left( 2a_n - a_1 - a_2 \right) \left( 2a_n - a_2 - a_3 \right) \cdots \left( 2a_n - a_{n-2} - a_{n-1} \right) \left( 2a_n - a_{n-1} - a_1 \right).$$

Let's think step by step and output the final answer within \boxed{}.
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