When Mayor Bill de Blasio announced a search for a new schools chief, he said he would look for someone very much like retiring schools Chancellor Carmen Fariña. But as the search intensifies, that approach could pose a dilemma for the mayor, and make it harder to find someone well-suited to running and improving a system that sees itself as a model for school districts across the country.

On the one hand, the search for someone in Fariña’s mold would make the position ideal for a proven leader from within the department. But some observers said there is a dearth of internal options, and that few of Fariña’s deputies have the right background or public persona to take on the role.

At the same time, looking outside the education department could present its own challenges, especially if the job will emphasize carrying out an existing agenda rather than encourage making key changes to a system that has made some incremental gains in recent years but still faces big inequities.

“If you’re just going to keep the trains running on time that’s not going to be a huge incentive to run an education system, even if it’s New York City,” said Josh Starr, a former schools superintendent in Connecticut and Maryland who was considered for the job four years ago. “The best person is probably someone who’s been a lifer in the [education department] who they trust.”

The appeal of running America’s largest school system — which is by itself the size of a large city — might seem obvious. It represents a chance to shape education policy in the national spotlight under a mayor who has shown an interest in ambitious education programs, including the widely-praised launch of free prekindergarten for every family.

But candidates might also be dissuaded by an arrangement where most major policy initiatives in recent years have been handed down from City Hall, there is a relatively short time frame to implement changes, and a job description that suggests the next chancellor will be tasked with continuing the existing blueprint rather than making his or her own mark.

Meanwhile, the next chancellor will have to tackle difficult problems that the current administration has struggled to address: how to serve a growing number of homeless students, yawning achievement gaps, and persistent requests from charter schools for more space to operate. On top of that, whoever takes the job might have less funding to tackle those issues.

“We’re in for some really horrible budget times with the new tax law and there’s going to be a lot of pressure on Washington to cut budgets,” said Clara Hemphill, director of education policy and Insideschools at the New School’s Center for New York City Affairs.

Still, whoever lands the job will inherit a school system that has enjoyed relative stability and progress: Test scores and graduation rates have incrementally climbed, and perhaps the mayor’s most ambitious education initiative — universal pre-K — has been widely hailed as a success. The relative calm means the next chancellor will be taking over a system many consider to be on the right track, though that could complicate efforts push through more ambitious — and potentially risky — changes.

In many ways, that legacy is exactly what de Blasio wanted from Fariña. In contrast with former Mayor Michael Bloomberg, who tried to systematically disrupt the system, Fariña was brought in to rebuild trust with educators and lend a steady hand. Some observers say she accomplished that mission — conducting legendary school visits to gauge the system’s health instead of leaning on reams of data, repairing the department’s relationship with the teachers union, and focusing on small-bore policies to encourage collaboration rather than competition between schools and educators.

But critics say the administration has never articulated a clear vision for improving the system as a whole, refused to tackle key equity issues like school segregation head on, and struggled with an expensive program to improve low-performing schools that has at best achieved mixed results — challenges a new chancellor must contend with.

Another legacy, some observers said, is that Fariña has not set up a natural successor. The New School’s Hemphill said that few of the department’s top deputies — including Dorita Gibson, Fariña’s second in command — have the visibility or political savvy to take on the job. And Josh Wallack, the deputy who helped execute the mayor’s signature pre-K expansion, does not meet de Blasio’s requirement that the chancellor must be an educator.

Finding an outsider to take the post could present challenges of its own. Not only is the chancellor’s salary is relatively low compared with some other school systems, which could give pause to the head of a higher-paying district, but also City Hall has engineered many of this administration’s high-profile education efforts.

Pedro Noguera, a UCLA professor who was reportedly floated for the role, said that the administration’s lack of a unifying vision for the system dissuaded him from considering the position.

“The items on their broad list – I have no objection to those,” he said in an interview. “But I don’t think they add up to the strategy that the system needs.” (Noguera said he was contacted by a third party about the job, but not directly by city officials.)

Still, many observers are optimistic about the mayor’s odds of fielding strong candidates. Fariña’s predecessor, Dennis Walcott, said running the nation’s largest school system will be difficult to resist — and there’s plenty of time for the next chancellor to make their mark.

“The person will be in charge of the largest municipal agency in the country,” he said. “The mayor is term limited, but he has a long four years in front of him.”