Completely hypothetical. What tools would a President of Party 1 have to accomplish work if the Senate/House were virtually all from Party 2? By work, I would mean work that aligns to their respective party's views.
Rather than going into a lengthy catalog of Article 2 powers and the attendant case law that shows an historical accretion of power in the Presidency, along with perhaps the growth of the Executive Department and administrative state (those are easily catalogued, but my college text on the matter ran over 2000 pages), I will turn to the resource modern scholars use, Richard Neustadt's, summarized as
The power to persuade is perhaps the most important aspect of the presidency that Neustadt writes about. The power of the United States government is vastly dispersed; the president cannot simply command and receive. Other levels of government have different constituencies and different sources of power and interest. The president is one man and needs others to get things done. The president must bargain and persuade others that what he wants is in their best interest. President Truman once said of President Eisenhower upon his election, "He'll sit there all day saying do this, do that, and nothing will happen. Poor Ike, it wont be a bit like the military. He'll find it very frustrating." Neustadt refers to the president in this respect, as a "clerk", in which the president must balance differing interests. Just because the president wants something done does not mean that the others who also possess the power and authority will carry out his wishes. " The presidents advantages are checked by the advantages of others. Relationships will pull in both directions. These are relationships of mutual dependence. The president depends upon the persons that he would persuade; he has to reckon with their need or fear of them (Neustadt 31)." The president must interpret to his colleagues how his policy will benefit them as well.
Neustadt then breaks this down a little further:
The president's resources include the bargaining powers that come with the position, professional reputation, and public prestige.
The president's professional reputation involves how others expect him to react. Isolated failures are not a problem, but if the failures form a pattern, this will weaken him. In addition to anticipating what the president wants, others also have to assess how hard he will try to get it. Tenacity is important. If a president cannot convince others that he will inevitably win, at least he needs to convince them that it will be costly to cross him. You can't punish everyone, but you need to selectively punish your enemies and reward your allies.
Public prestige deals with the president's popular support outside Washington. (With reputation, people anticipate the reactions of the president; with prestige, they anticipate the reactions of the voters.) Most politicians and bureaucrats do not watch poll numbers directly; they watch Congress. Prestige conveys leeway because low prestige encourages resistance.
The president must safeguard his power personally. No one else sees politics from the same vantage point, and so no one else can do this for him. Everyone else has the institutional pulls of their position tinting their judgment. "Yet nobody and nothing helps a President to see, save as he helps himself"
So ascendant is this view of the modern Presidency many will ascribe that persuasion is in fact the only power that a President commands.
Traditionally the senior leadership of each party would work together to pass bills some of which were suggested by the President. The President is usually only minimally involved in the process until it ends up at his desk. Prior to that its common for Presidents to make their wishes known as to what a bill must have or what is a deal breaker. For signature legislation the President may take a more active role in the process before it gets to his desk, such as with Obamacare.
A President facing a congress led by the opposing party generally compromises to some degree and "moves towards the center." Clinton was particularly notable in a President that accomplished a lot with an opposing congress, though most Presidents have compromised to some degree. This does require both the president and opposing party to be willing to work together which hasn't happened much at all since Clinton.
History says "yes," but the current political polarization in America suggests that the outcome might be otherwise.
When Reagan was elected, he faced a Democratic House, and a Senate that was nominally Republican (but included more genuine centrists than you would find today). Nevertheless, Reagan was able to re-orient the direction of American politics and policy in lasting and substantial ways.
Political effectiveness in the case of mixed party control depends on goodwill, and a willingness to compromise. That existed in America in 1981 in a way that it doesn't appear to now: on matters of deep policy substance, there are only a small handful of Congresspersons that are willing to cross party lines for a vote.
Tools that could be used to increase effectiveness would either require building and exploiting goodwill, motivating public support, or increasing willingness to compromise. Given the premise of your question (that virtually all legislators are of a differing party), the President would have to use every tool at his disposal (charm, pressure, horse trading, public opinion), as well as compromise his own preferences to a great degree.