Although the Soviet space effort is and was presented as a smooth running, monolithic example of Soviet efficiency and superior science, it was internally a chaotic mess, riven by rivalries and factionalism. To compound the problem, while the Soviets were initially able to get the jump on the Americans with Sputnik and the early manned space program, by the mid 1960's it was very apparent that Soviet rocketry had stalled. The "brute force" approaches of the early Soviet program may have developed the massive R-7 booster/ICBM, but American technology had made considerable progress in all kinds of areas, from miniaturized computers, the ability to reliably produce and handle high energy cryogenic fuels to fairly startling advances in rocket engine design. The F-1 rocket motors of the Saturn V are still unmatched for sheer size and power, and while the monster Saturn V launch stack used 5 of thee beasts to lift off (generating @ 35 GW of energy in the process) the comparable N-1 ganged 30 NK-15 engines for its first stage...
Saturn V compared to the N-1
The real problem isn't to be solved by Korolev or Chelomei via technical means, but much, much higher in the Soviet hierarchy. The lack of interest in space exploration by the Soviet Strategic Rocket forces and the mercurial temperament first of Khrushchev, then after his overthrow by the different priorities of Brezhnev, First Deputy Premier Alexander Shelepin, and KGB Chairman Vladimir Semichastny, meant the N-1 project started late and was always underfunded.
So, how to fix the problem is to look at one of the alternatives proposed in 1961. Korolev proposed a lunar mission based on what came to be known as Soyuz, assembling the lunar mission in Earth orbit using multiple launches. This program, while technically ambitious, does not require the massive leap in technology that the N-1 represented. By using "assembly line" production of the booster and many of the spacecraft parts, quality control is improved, and any launch failures can simply be replaced with another launch a short time later. The Strategic Rocket Forces may find this acceptable since high rate production of the new booster will allow them to quickly replace older R-7 generation ICBM's with the new rocket, or an ICBM largely derived from the new rocket.
Apollo CM compared to the Zond
The one critical piece in this plan was initially Korolev still needed an N-1 launch to bring the lunar lander and fuel into orbit, but you can speculate that with a high rate of production and launches, Soviet space experience and technical refinement might allow all the pieces to be lofted by existing boosters in the final configuration.
LK compared to the LEM
This is still a fairly primitive and cobbled together system, especially compared to the much larger and more capable Apollo CM and LEM, but it has the advantage of starting earlier, having the potential to attract more internal support from the Soviet hierarchy and allow the USSR to reach the Moon first.