# 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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