Strange Functional Equation

Algebra Level 3

\( f(x)\) is an even function with codomain \(\mathbb{R}.\) Its graph is symmetric about the line \( x=1 \), and \[f(x_{1}+x_{2})=f(x_{1})\cdot f(x_{2}) \] for any \( x_{1},x_{2}\in [0,\frac{1}{2}] \), and \( f(1)> 0. \)

Let \( a_{n}=f(2n+\frac{1}{2n}) \). Find the value of \( \displaystyle\lim_{n\rightarrow \infty }(\ln\: a_{n}) \).

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