All these answers are good and correct. I would add "to depend (upon/on) someone/something":
"I counted on (depended on) the tow line (tested to forces of 1200 newtons) to safely maneuver my yacht every day."
A VERY BROAD & GENERALIZED rule when learning English is that verbal phrases and prepositional phrases sound more natural, less formal and less "foreign/exotic/special" than a single-word verb. Examples:
verbal phrase vs. single verb
"to count on" = "to depend"
"to dig up" = "to excavate"
"to screw up" (very informal) = "to equivocate"
"to come in" = "to enter"
"to get up" = "to rise"
There are thousands more examples of this, but what I am saying is that "count on" is perfectly useful but less formal.