Climbing a staircase

Algebra Level 3

Let \( F(x) \) be a non-decreasing function defined on \( [0, 1 ] \) such that

\[ \begin{cases} 2 F \left ( \frac{x}{3} \right) = F(x), \\ F(x) + F(1-x) = 1 .\\ \end{cases} \]

What is the value of \( F ( \frac{1}{13} )\)?

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