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Gov. Dannel P. Malloy on Wednesday leaned away from legislation that would allow a doctor to prescribe lethal medication to a dying patient in favor of a policy giving patients broader rights to refuse treatment.

“It’s a very tricky issue,” the governor said at an unrelated press conference when he was asked about the policy, which has been labeled “aid-in-dying” by supporters and “assisted suicide” by critics.

The legislature is considering the topic this session for the second consecutive year. The Public Health Committee is holding a public hearing on the subject on Monday. Last year’s legislation lacked enough support to make it out of the committee.

But supporters are encouraged this year by a Quinnipiac University poll released last week that suggested 61 percent support the concept. Malloy said there is more than one proposal this year designed to increase options available to very sick patients.

“I’m not sure how clear some folks are about which one they actually support,” he said.

The governor endorsed another piece of legislation which would create a pilot program on “medical orders for life sustaining treatment.” The idea is similar to “do not resuscitate” orders, which are already in use in Connecticut. It would give seriously ill patients the option of creating directives indicating they want to refuse life-sustaining treatments.

“I support that legislation and I think people in Connecticut overwhelmingly support that legislation,” Malloy said.

The governor said his position on the bill allowing doctors to prescribe lethal doses of medication will depend upon the specific language lawmakers send him, if they pass the bill. But he made clear that his preference is for legislation on medical orders.

“If it’s more like the directive bill as opposed to an assisted suicide bill, then I think it will pass. If it’s an assisted suicide bill, I think it’s going to get some opposition,” he said. “I’ll work with folks to craft the best bill possible that will yield the citizens of our state the most say over their treatment, and what treatment would be undertaken to prolong life and what treatments under what conditions would not take place.”

Asked whether he supported assisted suicide policies that have been adopted in other states, Malloy said it depends on the state.

“I would oppose some of those and I might — they’re pretty broad . . . there’s a pretty big difference between some of those bills. So until I see an actual package, it’s tough for me to give you that,” he said. “I am 100 percent behind additional directive language and certainly will consider other language.”

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Tim Appleton, state director of Compassion and Choices Connecticut, said polls have consistently suggested strong public support for aid-in-dying policy. He said he was looking forward to Monday’s hearing in the Public Health Committee, where he hoped supporters would be able to win over some skeptics.

“It’s our hope that during that opportunity we can change the minds of some legislators and perhaps better inform the governor on this important issue, which most people in state of Connecticut view as a fundamental right,” he said.

Public Health Commissioner Jewel Mullen said she has not read the aid-in-dying bill, but she said the treatment directive policy, which Malloy supports, will offer some options to the terminally-ill patients who would be impacted.

“The legislation you’re asking me about is really designed to give people some autonomy in their choice about dying. This bill . . . is giving people more autonomy about treatments that they undergo while they’re living,” she said. “The M.O.L.S.T. is not a fast-track to death.”

Mullen said that under that bill, patients will have a tool to opt out of things like antibiotic treatment or the use of a feeding tube. The directives would be in patients’ records in case they were rendered unconscious or in a vegetative state.