In R conditional statement with lag value and starting value

I want to create a new column y that adds x to previous y value (i.e. lag(y)) and prints based on conditions below. The start value of y is 44169.

if (x + lag(y)) is greater or equal than 44169, than 44169 is printed.
if (x + lag(y)) is less than 44169, then print x + lag(y).

I am tripped up by the lag and how to handle the starting value. Data and desired results are below, Thanks for your help

date<-seq(as.Date('1968-04-01'),as.Date('1968-09-01'), by='month')
x<-c(20,10,-15,-12,5,-50)
df<-data.frame(date,x)

tried and failed

df2<- df %%
  mutate(y = if (x+lag(y) > 44169) {
      44169}
  else {x + lag(y)}
    )

y<-c(44169, 44169, 44154, 44142, 44147, 44169)
result<- data.frame(date,x,y)

We may use accumulate

library(dplyr)
library(purrr)
df %>% 
  mutate(y = accumulate(x, ~
     if((.x + .y) > 44169) 44169 else .x + .y, .init = 44169)[-1])

Or with pmin/min

df %>%
   mutate(y = accumulate(x, ~ pmin(44169, .x + .y), .init = 44169)[-1])

Dev solutions