SIRIUS Satellite Radio and XM Satellite Radio announced the completion of their merger, which will be called SIRIUS XM Radio Inc. XM shareholders will receive shares of SIRIUS common stock for each share of XM stock. The stock price of the acquiring company, Sirius, prior to the merger announcement was approximately $$3\mathrm{.}7$ per share, and SIRIUS had $1,460$ million shares outstanding. XM$,$ the target company, had a stock price around $$14$ and $314$ million shares outstanding. Assume the value of the synergies $($merger gain$)$ was expected to be $$1$ billion.

a$)$What is the minimum exchange ratio that XM shareholders would be willing to accept?

b$)$ What is the maximum exchange ratio that SIRIUS shareholders would be willing to accept?

c$)$ What is the exchange ratio that would allocate $30\%$ of the synergy gains to SIRIUS shareholders?