Question: Which event began first, the Western Allied invasion of Germany or the Battle of Aachen?

full_context: Document 1: Operation Nordwind: Operation North Wind (German: "Unternehmen Nordwind" ) was the last major German offensive of World War II on the Western Front. It began on 31 December 1944 in Alsace and Lorraine in northeastern France, and ended on 25 January.
Document 2: Battle of Hürtgen Forest: The Battle of Hürtgen Forest (German: "Schlacht im Hürtgenwald" ) was a series of fierce battles fought from 19 September to 16 December 1944 between American and German forces on the Western Front during World War II in the Hürtgen Forest about 50 mi2 east of the Belgian–German border. It was the longest battle on German ground during World War II, and is the longest single battle the U.S. Army has ever fought.
Document 3: Norwegian Campaign order of battle: The German operation for the invasion of Denmark and Norway in April 1940 was code-named "Weserübung", or "Weser Exercise." Opposing the invasion were the partially mobilized Norwegian military, and an allied expeditionary force composed of British, French, and Free Polish formations. The following list formed the order of battle for this campaign.
Document 4: Allied-occupied Germany: Upon the defeat of Nazi Germany in World War II, the victorious Allied powers asserted their joint authority and sovereignty over 'Germany as a whole', defined as all territories of the former German Reich which lay west of the Oder–Neisse line; having declared the extinction of Nazi Germany at the death of Adolf Hitler (see 1945 Berlin Declaration). The four powers divided 'Germany as a whole' into four occupation zones for administrative purposes, creating what became collectively known as Allied-occupied Germany (German: "Alliierten-besetztes Deutschland" ). This division was ratified at the Potsdam Conference (17 July to 2 August 1945). In autumn 1944 the United States, United Kingdom and Soviet Union had agreed on the zones by the London Protocol. The powers at Potsdam approved the detachment from 'Germany as a whole' of the German eastern territories east of the Oder-Neisse line; with the exact line of the boundary to be determined at a final German Peace Treaty. This treaty was expected to confirm the "shifting westward" of Poland's borders (back to approximately as they were before 1722), as the United Kingdom and the United States committed themselves to support there the permanent incorporation of former eastern German territories into Poland and the Soviet Union. In the closing weeks of fighting in Europe, United States forces had pushed beyond the agreed boundaries for the future zones of occupation, in some places by as much as 200 mi . The so-called line of contact between Soviet and American forces at the end of hostilities, mostly lying eastward of the July 1945-established inner German border was temporary. After two months in which they had held areas that had been assigned to the Soviet zone, U.S. forces withdrew in the first days of July 1945. Some have concluded that this was a crucial move that persuaded the Soviet Union to allow American, British and French forces into their designated sectors in Berlin, which occurred at roughly the same time (July 1945), although the need for intelligence gathering (see Operation Paperclip) may also have been a factor.
Document 5: 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 6: 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.
Document 7: Defence of the Reich: The Defence of the Reich (German: "Reichsverteidigung" ) is the name given to the strategic defensive aerial campaign fought by the "Luftwaffe" over German-occupied Europe and Nazi Germany during World War II. Its aim was to prevent the destruction of German civilians, military and civil industries by the Western Allies. The day and night air battles over Germany during the war involved thousands of aircraft, units and aerial engagements to counter the Allied strategic bombing campaign. The campaign was one of the longest in the history of aerial warfare and with the Battle of the Atlantic and the Allied Blockade of Germany was the longest of the war. The "Luftwaffe" fighter force defended the airspace of German-occupied territory against attack, first by RAF Bomber Command and then against the United States Army Air Forces (USAAF).
Document 8: 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 9: Battle of Kassel (1945): The Battle of Kassel was a four-day struggle between the U.S. Army and the German Army in April 1945 for Kassel, a medium-sized city 140 kilometers northeast of Frankfurt am Main, which also is the second-largest city in Hesse (after Frankfurt). The battle resulted as the U.S. Third Army pushed northeast from the region of Frankfurt and Mainz. The battle opened on April 1, 1945 and ended with an American victory three days later. Opposing the Third Army's 80th Infantry Division were an infantry replacement battalion, some heavy tanks, and anti-aircraft guns. Although the Germans gave battle at Kassel, their army was on the brink of collapse as the Western Allies and the Red Army made deep inroads into Germany. The defense of Kassel did not materially impede the Allied advance, and, one month after the battle ended, Germany was forced to capitulate.
Document 10: Battle of the Bulge: The Battle of the Bulge (16 December 1944 – 25 January 1945) was the last major German offensive campaign on the Western Front during World War II. It was launched through the densely forested Ardennes region of Wallonia in eastern Belgium, northeast France, and Luxembourg, towards the end of World War II. The surprise attack caught the Allied forces completely off guard. American forces bore the brunt of the attack and incurred their highest casualties of any operation during the war. The battle also severely depleted Germany's armored forces, and they were largely unable to replace them. German personnel and, later, Luftwaffe aircraft (in the concluding stages of the engagement) also sustained heavy losses.
Document 11: Battle of Arracourt: The Battle of Arracourt was a major clash between U.S. and German armored forces near the town of Arracourt, Lorraine, France, between 18–29 September 1944, during World War II. As part of a counteroffensive against recent U.S. advances in France, the German 5th Panzer Army had as its objective the recapture of Lunéville and the elimination of the U.S. XII Corps bridgehead over the Moselle River at Dieulouard. With local superiority in troops and tanks, the Germans anticipated quick defeat of the defending Combat Command A (CCA) of the U.S. 4th Armored Division. However, due in part to poor German tactics and the terrain, the 4th Armored Division's CCA, in concert with U.S. tactical air forces, defeated two Panzer Brigades and elements of two "Panzer" divisions in a series of engagements over an eleven-day period.
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: Battle of Nuremberg (1945): The Battle of Nuremberg was a five-day battle between the forces of the United States 7th Army on one side, and Nazi Germany and Russian Liberation Army volunteers on the other during World War II. The battle saw some of the fiercest urban combat during the war and it took four days for the United States to capture the city. The battle was a blow to Nazi Germany as Nuremberg was a center of the Nazi regime. Many rallies took place in the city and to lose the city to the Americans took a heavy toll on already low German morale. Even though American forces heavily outnumbered the German forces, it wasn't until 20 April, that the 7th Army took the city center. The battle devastated the city.
Document 14: 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 15: Battle of Aachen: The Battle of Aachen was a major combat action of World War II, fought by American and German forces in and around Aachen, Germany, between 2–21 October 1944. The city had been incorporated into the Siegfried Line, the main defensive network on Germany's western border; the Allies had hoped to capture it quickly and advance into the industrialized Ruhr Basin. Although most of Aachen's civilian population was evacuated before the battle began, much of the city was destroyed and both sides suffered heavy losses. It was one of the largest urban battles fought by U.S. forces in World War II, and the first city on German soil to be captured by the Allies. The battle ended with a German surrender, but their tenacious defense significantly disrupted Allied plans for the advance into Germany.
ground_truth: The Western Allied invasion of Germany