Question: which town in Rensselaer County, New York, United States did  Valley Falls, New York village lies on the boundary

Context paragraphs:
Seneca Falls (CDP), New York: Seneca Falls is a hamlet (and census-designated place) in Seneca County, New York, in the United States.  The population was 6,681 at the 2010 census.  The hamlet is in the Town of Seneca Falls, east of Geneva.  It was an incorporated village from 1831 to 2011.  Seneca Falls became the largest New York village ever to approve dissolution after village residents voted to dissolve it in 2010.

Josiah Masters: Josiah Masters (November 22, 1763 – June 30, 1822) was a United States Representative from New York.  Born in Woodbury, Litchfield County, Connecticut, he was graduated from Yale College in 1783.  He studied law, was admitted to the bar and commenced practice in Schaghticoke, Rensselaer County, New York.  He was a member of the New York State Assembly in 1792, 1800, and 1801, and served as supervisor of Schaghticoke in 1796.  He was a justice of the peace in Rensselaer County from 1801 to 1805, was a trustee of Lansingburgh Academy, and was school commissioner of Schaghticoke.

Adams–Myers–Bryan Farmstead: Adams–Myers–Bryan Farmstead is a historic home and farm and national historic district located at Valley Falls, Rensselaer County, New York.  The farmhouse was built about 1855, and consists of three blocks.  It consists of a two-story, Greek Revival style main block with a two-story side wing and 1 1/2-story rear ell.  Also on the property are the contributing main barn group (c. 1860, c. 1880-1930), pump house (c. 1920-1940), milk house (c. 1920-1940), horse barn (c. 1850), corn house (c. 1850), pig house (mid-19th century), carriage barn (c. 1860-1875), ice house (c. 1900), and outhouse (c. 1900).

New York State Route 151: New York State Route 151 (NY 151) is an 8.23 mi state highway located in Rensselaer County, New York, in the United States.  Its western terminus is at an interchange with U.S. Route 9 (US 9) and US 20 in Rensselaer, and its eastern terminus is at a junction with NY 150 in Shivers Corners, a hamlet in the town of Schodack.  It also has an intersection with US 4 in East Greenbush.  NY 151 was assigned to most of its current alignment as part of the 1930 renumbering of state highways in New York.  In Rensselaer, NY 151 has been altered several times, with the most recent change coming in the late 1990s.  The current western terminus of NY 151 was once the western terminus of NY 43.

Ridgefield Township, New Jersey: Ridgefield Township was a township that existed in Bergen County, New Jersey.  The township was created in 1871, when Hackensack Township was trisected to form Palisades Township in the northernmost third, Englewood Township in the central strip and Ridgefield Township encompassing the southernmost portion, stretching from the Hudson River on the east to the Hackensack River, with Hudson County to the south.  Much of the area had been during the colonial area known as the English Neighborhood.  As described in the 1882 book, "History of Bergen and Passaic counties, New Jersey," "Ridgefield is the first township in Bergen County which the traveler enters in passing up the Palisades.  His first impressions are much like those of old Hendrick Hudson in speaking of a wider extent of country: "A very good land to fall in with, and a pleasant land to see."  The valley of the Hackensack invited early settlers in the seventeenth century, and the valley of the Overpeck Creek, a navigable arm of the Hackensack, also attracted settlers quite as early in this direction.  Sloops and schooners can pass up this creek nearly to the northern boundary of the township.  Ridgefield is bounded on the north by Englewood, on the east by the Hudson, on the south by Hudson County, and on the west by the Hackensack River.  The southern boundary is less than two miles in extent, and the northern less than four, and the length of the township from north to south does not exceed four miles.  Bellman's Creek, forming part of the southern boundary, the Hackensack, the Overpeck, the Hudson, with more than a dozen other smaller streams and rivulets, bountifully supply the whole township with water.  From the western border of the Palisades the land descends to the Overpeck, forming a most beautiful valley, with the land again rising to a high ridge midway between the Overpeck and the Hackensack.  From this long ridge, extending far to the north beyond this township, it took its name of Ridgefield. <br><br>The New York, Susquehanna and Western, formerly the Midland Railroad, the Jersey City and Albany Railroad, and the Northern Railway of New Jersey—all running northward through the township— afford ample railroad accommodations.  The Susquehanna enters the township at Bellman's Creek, and the Northern at about one hundred feet south of the creek, and at a point north and east of the Susquehanna.  The Albany road in this locality is not yet constructed, diverging at present from the track of the Susquehanna between Little Ferry and Bogota stations.  It has, however, an independent line projected and now under construction to New York City. <br><br>Early Settlements.  Ridgefield embraces the earliest settlements in the ancient township of Hackensack, antedating even the organization of that township in 1693, and of the county of Bergen in 1675.  There seems to have been no town or village compactly built, like the village of Bergen, but there were settlements both of Dutch and English in and about what was subsequently known as English Neighborhood prior to 1675.  The Westervelts, the Zimcrmans, the Bantas, and the Blauvelts, all coming from Holland, settled in the middle of the seventeenth century in that locality.  The ancestors of Jacob P. Westervelt, now of Hackensack Village, with himself, were born in English Neighborhood.  His father was born there in 1776, and was the son of Christopher Westervelt, who was born there certainly as early as 1690, and he was the son of the original ancestor of this family, who came from Holland and settled on Overpeck Creek, within the present limits of Ridgefield township, probably about 1670."

New York State Route 154: New York State Route 154 (NY 154) was a state highway in Rensselaer County, New York, in the United States.  It began at an intersection with U.S. Route 4 (US 4) in the city of Troy and ended at then-County Route 77 (CR 77, now NY 351) in the hamlet of Poestenkill.  Along the way, it intersected NY 66 and NY 355.  The route was assigned in the early 1930s and eliminated in 1980 as part of a highway maintenance swap between the state of New York and Rensselaer County.  The portion of the highway between the Troy city line and NY 355 became County Route 130 while the part connecting NY 355 to Poestenkill became an extension of NY 355.

Valley Falls, New York: Valley Falls is a village in Rensselaer County, New York, United States.  The population was 466 at the 2010 census.  The village lies on the boundary of the towns of Pittstown and Schaghticoke, but is mostly in the northwest part of Pittstown.

Schaghticoke (town), New York: Schaghticoke is a town in Rensselaer County, New York, United States.  The population was 7,679 at the 2010 census.  It was named for the Schaghticoke, a Native American tribe formed in the seventeenth century from an amalgamation of remnant peoples of eastern New York and New England.  The tribe has one of the oldest reservations in the United States, located in what is now Litchfield County, Connecticut.  It has been recognized by the state of Connecticut but has not yet achieved federal recognition.

New York State Route 405: New York State Route 405 (NY 405) was a state highway in Rensselaer County, New York, in the United States.  It ran for 2.63 mi between an intersection with U.S. Route 4 in North Greenbush and a junction with NY 66 just inside the Troy city limits.  In between, NY 405 intersected NY 136.  The entirety of NY 405 was originally part of NY 40.  In the February–March 1973, NY 40 was truncated northward to NY 7 in northern Troy and its former routing from US 4 to NY 66 was assigned NY 405.  Ownership and maintenance of most of NY 405 was transferred to Rensselaer County on April 1, 1980, at which time the parts of the route given to the county were redesignated as part of County Route 74 (CR 74).

New York State Route 9J: New York State Route 9J (NY 9J) is a north–south state highway in the Hudson Valley region of New York in the United States.  It begins at an intersection with US 9 in the Columbia County town of Stockport and extends for 22.35 mi to an interchange with US 9 and US 20 in the Rensselaer County city of Rensselaer.  The route parallels the Hudson River for its entire length, and several parts of the highway run directly alongside the river.  NY 9J was assigned to its current alignment as part of the 1930 renumbering of state highways in New York.

Supporting paragraphs: Valley Falls, New York Schaghticoke (town), New York
Answer: Schaghticoke