Fiat money isn't actually valued at what people believe it to be. There's an actual way to determine value for currency not based on a gold or silver standard.
Fiat money is basically a promise of future work or substance. What determines the value is how secure a promise that is. Therefore, it's basically valued based on a country's level of "credit" for lack of a better term. This is similar to a gold or silver standard, as notes following either of the aforementioned standards have a solid, concrete source of "credit" to give the notes value, as the promise of substance is already backed up. However, their value is limited to the size of the source of gold or silver. However, fiat money allows money to basically be produced out of nothing. Work is a resource that is limitless in supply, so fiat money has an infinite-sized source to base their value off of. This is both a blessing and a curse. It allows for unparalleled growth compared to gold or silver standard currencies, but it also gives a certain degree of riskiness that isn't associated with them. Since the value of fiat currency does not have an accurate "value" to anchor it, it's value will fluctuate based on how much faith people are willing to put into the promise that the currency provides (basically how credit works). This means the value of fiat currency is at greater risk of crashing compared to quantitative standard currencies.
That being said, currency, in theory, should not be able to affect the value of goods. Currency, for all intents and purposes, really is just an arbitrary measurement to access worth. By measuring an object's length, do you change the length in the process? The obvious answer is no.
However, that's in theory. Since currency is a measurement to access worth, a qualitative rather than a quantitative value, it has the ability to influence the worth of the object it's measuring. An example would be that more expensive wine is perceived as tasting better and being worth more than cheaper wine, even if it's really the exact same wine just given different prices. So while the goods are identical, by giving one of them a higher price, you give the impression that the worth of the object is somehow greater, and actually make the worth of the object increase in the process. At the same time, lowering the prices of objects can actually increase the perceived value because of the value of saving money.
As a real world example, try googling Tulip Mania, or the Great Tulip Crash. Basically, and I shit you not, in the 1600s, Holland managed to crash their economy because the perception of the value of tulips got into a massive circlejerk with currency values and resulted in a massive bubble in the tulip bulb market, which inevitably resulted in a crash when somebody finally said, "Why the fuck am I paying 2 years salary for a fucking flower?" And not just any 2 years salary mind you. Wealthy business men 2 year salaries. And the flowers died in 2 years anyways. But for their credit, it was supposedly a very pretty tulip. Anyways, there's a real-life example for how currency and tulips fucked up a country's economy.
It's also worth mentioning that everything below the value of a quarter produced by the United States is not worth producing. There's a precedent for this as well. America had discontinued the half-pence a while back. If you count for inflation, the value of the half-pence comes out to around the value of a dime nowadays. And American military bases supposedly have done away with all denominations below 25 cents.