Super Powers

Calculus Level pending

Let \( {\left( a_n\right)}_{n \in \mathbb{N} } \) be a sequence defined, recursively, by:

\( \begin{cases} a_1=x \\ \forall n\in \mathbb{N}\backslash\{1\} , & a_{n+1}=x^{a_n}\\ \end{cases} \)

For some \( x \in \mathbb{R}^{+} \).

Let \( x\in \mathbb{R}^{+} \) be such that \( \lim a_n=2 \)

(these means, intuitively, that \( x^{x^{x^{x^{x^{ ... }}}}} =2\) ).

Find the value of \( \lfloor x^6+x^4+x^2+1 \rfloor \).

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