Suppose the three vectors \( \vec{a}, b, \vec{c} \) on a plane satisfy the condition that \( |\v...

Suppose the three vectors \( \vec{a}, b, \vec{c} \) on a plane satisfy the condition that \( |\vec{a}|=|\vec{b}|=|\vec{c}|=|\vec{a}+\vec{b}|=1 ; \vec{c} \) is perpendicular to \( \vec{a} \) and \( \vec{b} \cdot \vec{c}0 \), then for real numbers \( x, y \) the vector \( \vec{p}=x \vec{a}+y \vec{c} \) satisfies the condition \( 0 \leq \vec{p} \cdot \vec{a} \leq 1 \) and \( 0 \leq \vec{p} \cdot \vec{b} \leq 1 \). Find the maximum value of \( \vec{p} \cdot \vec{c} \).