By Tim Kelly and Nobuhiro Kubo

TOKYO (Reuters) - Rattled by North Korean military advances, influential Japanese lawmakers are pushing harder for Japan to develop the ability to strike preemptively at the missile facilities of its nuclear-armed neighbor.

Japan has so far avoided taking the controversial and costly step of acquiring bombers or weapons such as cruise missiles with enough range to strike other countries, relying instead on its U.S. ally to take the fight to its enemies.

But the growing threat posed by Pyongyang, including Monday's simultaneous launch of four rockets, is adding weight to an argument that aiming for the archer rather than his arrows is a more effective defense.

"If bombers attacked us or warships bombarded us, we would fire back. Striking a country lobbing missiles at us is no different," said Itsunori Onodera, a former defense minister who heads a ruling Liberal Democratic Party committee looking at how Japan can defend against the North Korean missile threat. "Technology has advanced and the nature of conflict has changed."

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For decades, Japan has been stretching the limits of its post-war, pacifist constitution. Successive governments have said Tokyo has the right to attack enemy bases overseas when the enemy's intention to attack Japan is evident, the threat is imminent and there are no other defense options.

But while previous administrations shied away from acquiring the hardware to do so, Prime Minister Shinzo Abe's LDP has been urging him to consider the step.

"It is time we acquired the capability," said Hiroshi Imazu, the chairman of the LDP's policy council on security. "I don’t know whether that would be with ballistic missiles, cruise missiles or even the F-35 (fighter bomber), but without a deterrence North Korea will see us as weak."

The idea has faced stiff resistance in the past but the latest round of North Korean tests means Japan may move more swiftly to enact a tougher defense policy.

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“We have already done the ground work on how we could acquire a strike capability,” said a source with knowledge of Japan's military planning. He asked not to be identified because of the sensitivity of the issue.

Any weapon Japan acquired with the reach to hit North Korea would also put parts of China's eastern seaboard within range of Japanese munitions for the first time. That would likely anger Beijing, which is strongly protesting the deployment of the advanced U.S. Terminal High Altitude Area Defense (THAAD) anti-missile system in South Korea.

"China has missiles that can hit Japan, so any complaints it may have are not likely to garner much sympathy in the international community," said Onodera. GROWING THREATS

Currently, more than three missiles at one would be too many for Japan's already stretched ballistic missile defense to cope with, another source familiar with Japan’s capability said.

One serious concern for Japan is North Korea's development of solid fuel systems demonstrated last month that will allow it to conceal preparations for missile strikes because it no longer needs fuel its missiles just prior to firing.

That test also demonstrated a cold launch, with the rocket ejected from its launcher before engine ignition, minimizing damage to the mobile launch pads. Japanese officials also noted that the launch truck was equipped with tracks rather than wheels, allowing it to hide off road.

North Korea says its weapons are needed to defend against the threat of attack from the United States and South Korea, which it is still technically at war with.

Japan is already improving its ballistic missile defenses with longer-range, more accurate sea-based missiles on Aegis destroyers in the Sea of Japan and from next month will start a $1 billion upgrade of its ground-based PAC-3 Patriot batteries.

Also under consideration is a land-based version of the Aegis system or the THAAD system.

Those changes, however, will take years to complete and may not be enough to keep pace with rocket technology advances by Pyongyang, the sources said.

A quicker option would be for Japan to deploy ground-to-ground missiles to defend against an attack on its Yonaguni island near Taiwan fired from bases on Japanese territory several hundred kilometers to the east. A missile with that range could also hit sites in North Korea.

Japan could also buy precision air launched missiles such as Lockheed Martin Corp's extended-range Joint Air-to-Surface Standoff Missile (JASSM) or the shorter-range Joint Strike missile designed by Norway's Kongsberg Defence Aerospace AS for the F-35 fighter jet.

But with limited capability to track mobile launchers, some Japanese officials still fear any strike would leave North Korea with enough rockets to retaliate with a mass attack.

"A strike could be justified as self defense, but we have to consider the response that could provoke," said another LDP lawmaker, who asked not to be identified. (This story was refiled to correct full name of THAAD in paragraph 10) (Editing by Lincoln Feast)