(Andrew Harrer/BLOOMBERG)

As the curtain raises on Act Two of the debt-reduction saga playing out on Capitol Hill — the appointment of a “supercommittee” to tackle the country’s toughest spending issues — a bipartisan path forward appears even more elusive than it did during Act One — the deal President Obama signed into law Tuesday to raise the country’s $14.3 trillion borrowing limit through the end of next year.

Rather than agree to a general framework and a ballpark figure for spending cuts equal to the amount of the debt-ceiling increase as they did during Act One, party leaders in Act Two will have to tackle what Vice President Biden has called the “philosophically big-ticket items”: key issues on which the parties have major differences, such as entitlement spending and tax reform.

Leaders of both parties have vowed to enter the next stage of the talks in a spirit of bipartisanship. But they’ve also provided for a “trigger” mechanism that would enforce across-the-board cuts in case the committee deadlocks — and the recent track record of committees would suggest that outcome is more likely than not.

When it comes to the next phase of the debt battle, the composition of the 12-member committee will be key. So why not start things off on in the most bipartisan of ways -- by having each party’s leaders in each chamber appoint the other side’s committee members?

The pluses of such an approach:

1) It would take internal party politics out of the equation, since leaders — who are already being lobbied by members who would like to serve on the committee — would not be pressured to appoint particular members of their own caucus for political reasons.

2) It could lead the parties to take on their “sacred cows,” since Republican leaders might appoint moderate Democrats open to entitlement reform, while Democratic leaders might appoint moderate Republicans willing to include revenue increases in a broader deal.

3) It could smooth the way for rank-and-file members to accept the sacrifices that any eventual debt-reduction deal will entail by forcing each party to make a sacrifice at the outset — thereby tamping down on members’ expectation that a sweeping debt deal will be painless.

4) It would make it harder for each party to blame the other for a potential stalemate, since each side’s leaders will have been responsible for choosing the other’s committee members.

Of course, there are potential downsides as well:

1) Rank-and-file members could revolt against their leadership if, as would likely be the case, the other side’s leaders picked moderates who agreed to a plan unpalatable to each party’s base.

2) Leaders could end up tapping members from the other side who might bring fresh voices and be closer to their own ideological views but who lack the budgetary chops to put together a serious fiscal plan.

3) One side or the other could decide that the committee idea is a terrible one and decide to sabotage the panel, picking members who they know will butt heads with each other.

4) One (or both) parties could try to use the committee for political purposes, selecting from the other side vulnerable members — or those with higher ambitions — in a move to thwart their electoral future.

What’s your take on the supercommittee? Who would end up on the panel if each side chose the other’s appointees? How about if members of each caucus voted on candidates among themselves?

Let’s hear your thoughts in the comments section below.