How is the word "gather/bring into refuge" related to the root עז?

QUESTION: Why does root עז or עוז mean to gather or bring in to a refuge, (Ex 9:19 & Isaiah 10:31), or to seek refuge (Isa. 30:10)? And is this root related to goats עזים, bold/fierce/strong עז, or other words that begin with עז, like עזר and עזב (assist and refuge seem to share common idea)? And how about עזק (Isa. 5:2), in sense of encircling and fencing in (like עזקא ring in Aramaic)? Refuges are usually encircled and enclosed, aren’t they? While we’re at it, what about לעז?