How is $1-$propoxypropane synthesized from propan$-1-$ol? Write the mechanism of this reaction.

(N/A) $1-$propoxypropane can be synthesized from propan$-1-$ol by acid-catalyzed dehydration. Propan$-1-$ol undergoes dehydration in the presence of protic acids (such as $H_{2}SO_{4}$ or $H_{3}PO_{4}$) at $413 \ K$ to give $1-$propoxypropane.
$2CH_{3}CH_{2}CH_{2}OH \xrightarrow{H^{+}, 413K} CH_{3}CH_{2}CH_{2}-O-CH_{2}CH_{2}CH_{3} + H_{2}O$
The mechanism of this reaction involves the following three steps:
Step $1$: Protonation
$CH_{3}CH_{2}CH_{2}-\ddot{O}H + H^{+} \rightleftharpoons CH_{3}CH_{2}CH_{2}-\overset{+}{O}H_{2}$
Step $2$: Nucleophilic attack
$CH_{3}CH_{2}CH_{2}-\ddot{O}H + CH_{3}CH_{2}CH_{2}-\overset{+}{O}H_{2} \to CH_{3}CH_{2}CH_{2}-\overset{+}{O}(H)-CH_{2}CH_{2}CH_{3} + H_{2}O$
Step $3$: Deprotonation
$CH_{3}CH_{2}CH_{2}-\overset{+}{O}(H)-CH_{2}CH_{2}CH_{3} \to CH_{3}CH_{2}CH_{2}-O-CH_{2}CH_{2}CH_{3} + H^{+}$