Question: Which took place longer in 1945, Battle of Luzon or Operation Undertone?

full_context: Document 1: Battle of Okinawa: The Battle of Okinawa (Japanese: 沖縄戦 , Hepburn: Okinawa-sen ) (Okinawan: 沖縄戦 , "Uchinaa ikusa " ), codenamed Operation Iceberg, was a major battle of the Pacific War fought on the island of Okinawa by United States Marine and Army forces against the Imperial Japanese Army and included the largest amphibious assault in the Pacific War during World War II, the April 1, 1945, invasion of Okinawa itself. The 82-day battle lasted from April 1 until June 22, 1945. After a long campaign of island hopping, the Allies were planning to use Okinawa, a large island only 340 mi away from mainland Japan, as a base for air operations for the planned invasion of the Japanese home islands.
Document 2: Operation Overlord: Operation Overlord was the codename for the Battle of Normandy, the Allied operation that launched the successful invasion of German-occupied Western Europe during World War II. The operation was launched on 6 June 1944 with the Normandy landings (Operation Neptune, commonly known as D-Day). A 1,200-plane airborne assault preceded an amphibious assault involving more than 5,000 vessels. Nearly 160,000 troops crossed the English Channel on 6 June, and more than two million Allied troops were in France by the end of August.
Document 3: Battle of Bataan: The Battle of Bataan (7 January – 9 April 1942) represented the most intense phase of Imperial Japan's invasion of the Philippines during World War II. In January 1942, forces of the Imperial Japanese Army and Navy invaded Luzon along with several islands in the Philippine Archipelago after the bombing of the American naval base at Pearl Harbor. The commander-in-chief of all Filipino and American forces in the islands, General Douglas MacArthur, consolidated all of his Luzon-based units on the Bataan Peninsula to fight against the Japanese invaders. By this time, the Japanese controlled nearly all of Southeast Asia. The Bataan peninsula and the island of Corregidor were the only remaining Allied strongholds in the region. Despite a lack of supplies, Filipino and American forces managed to fight the Japanese for three months, engaging them initially in a fighting retreat southward. As the combined Filipino and American forces made a last stand, the delay cost the Japanese valuable time and prevented immediate victory across the Pacific. The surrender at Bataan, with 76,000 soldiers surrendering in the Philippines altogether, was the largest in American and Filipino military histories, and was the largest United States surrender since the American Civil War's Battle of Harper's Ferry. Soon afterwards, Filipino and American prisoners of war were forced into the Bataan Death March.
Document 4: Battle of Luzon: The Battle of Luzon, fought 9 January – 15 August 1945, was a land battle of the Pacific Theater of Operations of World War II by the Allied forces of the U.S., its colony the Philippines, and allies against forces of the Empire of Japan. The battle resulted in a U.S. and Filipino victory. The Allies had taken control of all strategically and economically important locations of Luzon by March 1945, although pockets of Japanese resistance held out in the mountains until the unconditional surrender of Japan. While not the highest in U.S. casualties, it is the highest net casualty battle U.S. forces fought in World War II, with 192,000 to 205,000 Japanese combatants dead (mostly from starvation and disease), 10,000 American combatants killed, and between 120,000 to 140,000 Filipino civilians and combatants killed.
Document 5: Battle of the Visayas: The Battle of the Visayas was fought by U.S. forces and Filipino guerrillas against the Japanese from 18 March – 30 July 1945, in a series of actions officially designated as Operations Victor I and II, and part of the campaign for the liberation of the Philippines during World War II. The battle was waged to complete the recapture of the central portions south of the archipelago and secure them from remaining Japanese forces.
Document 6: Western Allied invasion of Germany: The Western Allied invasion of Germany was coordinated by the Western Allies during the final months of hostilities in the European theatre of World War II. The Allied invasion of Germany started with the Western Allies crossing the Rhine River in March 1945 before fanning out and overrunning all of western Germany from the Baltic in the north to Austria in the south before the Germans surrendered on 8 May 1945. This is known as the "Central Europe Campaign" in United States military histories.
Document 7: Battle of Baguio (1945): The Battle of Baguio occurred between 21 February and 26 April 1945, and was part of the greater Luzon campaign during the Allied liberation of the Philippines at the end of World War II. During the battle, American and Philippine forces recaptured the city of Baguio on the island of Luzon from a Japanese occupation force. One of the last tank engagements of the Philippine campaign took place during the battle. Baguio later became the scene of the final surrender of Japanese forces in the Philippines in September 1945.
Document 8: Battle of Bataan (1945): The Battle for the Recapture of Bataan from 31 January to 21 February 1945, by US forces and Allied Filipino guerrillas from the Japanese, part of the campaign for the liberation of the Philippines, was waged to secure the western shore of Manila Bay to enable the use of its harbor and open new supply lines for American troops engaged in the crucial battle for the liberation of Manila.
Document 9: South German Offensive: The South German Offensive is the general name of one of the final offensives of World War II in Europe. The offensive was led by the Seventh and Third armies of the United States along with the First Army of France. Soviet troops linked up with American forces in Czechoslovakia notably in the Battle of Slivice. The offensive was made by the US 6th Army Group to protect the US 12th Army Group's right flank and to prevent a German last stand in the Alps. However German resistance was much more fierce than in the north, which slowed the 6th Army Group's progress. However, by the end of April, many German divisions surrendered without a fight to the advancing American forces to avoid the inevitable destruction. The VI Corps of the Seventh Army linked up with the US Fifth Army, which fought through Italy, in the Alps as the Third Army advanced into Austria and Czechoslovakia, where it linked up with Soviet forces advancing from the east. Fighting continued a few days after the Surrender of Germany on 8 May, due to German forces fighting west to surrender to the Americans instead of the Soviets.
Document 10: Battle of Mindanao: The Battle of Mindanao was fought by United States forces and allied Filipino guerrillas against the Japanese from 10 March - 15 August 1945 on the island of Mindanao in the Philippines in a series of actions officially designated as Operation VICTOR V. It was part of the campaign to liberate the Philippines during World War II. The battle was waged to complete the recapture of the southernmost portions of the archipelago.
Document 11: Operation Undertone: Operation Undertone was a large assault by the U.S. Seventh and French 1st Armies of the U.S. Sixth Army Group as part of the Allied invasion of Germany in March 1945 during World War II.
Document 12: Spring 1945 offensive in Italy: The Spring 1945 offensive in Italy, codenamed Operation Grapeshot, was the final Allied attack during the Italian Campaign in the final stages of the Second World War. The attack into the Lombardy Plain by the 15th Allied Army Group started on 6 April 1945, ending on 2 May with the formal surrender of German forces in Italy.
Document 13: Sixth United States Army Group: The Sixth United States Army Group was an Allied Army Group that fought in the European Theater of Operations during World War II. Made up of field armies from both the United States Army and the French Army, it fought in France, Germany, Austria, and, briefly, Italy. Also referred to as the Southern Group of Armies, it was established in July 1944 and commanded throughout its duration by General Jacob L. Devers.
Document 14: Battle of the Philippine Sea: The Battle of the Philippine Sea (June 19–20, 1944) was a major naval battle of World War II that eliminated the Imperial Japanese Navy's ability to conduct large-scale carrier actions. It took place during the United States' amphibious invasion of the Mariana Islands during the Pacific War. The battle was the last of five major "carrier-versus-carrier" engagements between American and Japanese naval forces, and pitted elements of the United States Navy's Fifth Fleet against ships and aircraft of the Imperial Japanese Navy's Mobile Fleet and nearby island garrisons. This was the largest carrier-to-carrier battle in history.
Document 15: Operation Veritable: Operation "Veritable" (also known as the Battle of the Reichswald) was the northern part of an Allied pincer movement that took place between 8 February and 11 March 1945 during the final stages of the Second World War. The operation was conducted by Field Marshal Bernard Montgomery's Anglo-Canadian 21st Army Group, primarily consisting of the First Canadian Army under Lieutenant-General Harry Crerar and the British XXX Corps under Lieutenant-General Brian Horrocks. The U.S. Ninth Army was incorporated into the 21st Army Group. The objective of the operation was to clear German forces from the area between the Rhine and Maas rivers, east of the German/Dutch frontier, in the Rhineland. It was part of General Dwight D. Eisenhower's "broad front" strategy to occupy the entire west bank of the Rhine before its crossing. "Veritable" (originally called "Valediction") had been planned for execution in early January, 1945 when the ground had been frozen and thus more advantageous to the Allies. The Allied expectation was that the northern end of the Siegfried Line was less well defended than elsewhere and an outflanking movement around the line was possible and would allow an early assault against the industrial Ruhr region.
ground_truth: Battle of Luzon