Uranium$-238($U$-238)$ has a half$-$life of $4\mathrm{.}5$ billion years. Geologists find a rock containing a mixture of U$-238$ and lead, and determine that $77\%$ of the original U$-238$ remains; the other $23\%$ has decayed into lead. How old is the rock?

The rock is $$ billion years old.

$($Round to three decimal places as needed.$)$