Double Dip or Switch Satchels

Two identical velvet satchels sit on a table. Satchel one holds four $\$1$ chips and six $\$10$ chips; satchel two holds three $\$1$ chips and seven $\$10$ chips — you cannot tell which is which. You pick a satchel at random and draw one chip: it is a $\$1$. Keeping that chip out, you may draw and keep one more chip from either satchel. Which satchel should you draw from, and what is the expected value of that second draw?