How to make input$action starts from zero every time I click on Action Buttom in Shiny?

It may be simpler to not use the value of the actionButton directly and just record the number of clicks between events in an external reactiveValues object. This for example has two buttons, one to increase the value and another to add the final value to a vector and reset the counter:

library(shiny)


ui <- fluidPage(
  actionButton("action", label = "Action"),
  actionButton("store", label = "Store"),
  
  hr(),
  fluidRow(column(2, verbatimTextOutput("value")))
)

server <- function(input, output) {
  
  values <- reactiveValues(curr = 0,
                         out = integer())
  
  observeEvent(input$action, {
    values$curr = values$curr + 1
  })
  
  observeEvent(input$store, {
    values$out <- c(values$out, isolate(values$curr))
    values$curr <- 0
  })
  
  output$value <- renderPrint({
    values$out
    })
}

shinyApp(ui, server)

^{Created on 2022-01-06 by the reprex package (v2.0.1)}

If we really need to "reset" the button, we can re-render it. Here is an example app:

library(shiny)
library(dplyr)

ui <- fluidPage(
  uiOutput("action"),
  actionButton("reset", "Reset"),
  verbatimTextOutput("values")
)

server <- function(input, output, session) {
  output$action <- renderUI({
    actionButton("action_but", "Action")
  })

  vals <- reactiveVal(tibble(clicks = integer(0)))
  counter <- reactiveVal(0)

  observe({
    counter(input$action_but)
  })

  observeEvent(input$reset,
    {
      vals(add_row(vals(), clicks = as.integer(input$action_but)))

      output$action <- renderUI({
        actionButton("action_but", "Action")
      })

      counter(input$action_but)
    },
    ignoreInit = TRUE
  )

  output$values <- renderPrint({
    list(counter(), vals())
  })
}

shinyApp(ui, server)

The input$button is read only, so you can't reset the input$button increment without resetting the entire app, which would cause you to lose your saved values. Alternatively, you can keep a copy of the values and reset the count on your copy.

Not sure what you want to do with the reset, but you can still count the number of button presses between your steps, 3, 8 etc. So 8 button presses from 3 would be at input$button == 11.

library(shiny)

ui <- fluidPage(
  
  # Copy the line below to make an action button
  actionButton("action", label = "Action"),
  
  hr(),
  fluidRow(column(2, verbatimTextOutput("value")),
           column(6, verbatimTextOutput("countOutput")))
  
)


server <- function(input, output) {
  
  # You can access the value of the widget with input$action, e.g.
  output$value <- renderPrint({ input$action })
  
  df <- reactiveValues(three = NULL, eight = NULL, thirteen = NULL)
  
  observe({
  
    presses <- c(3, 8, 16)
      
    if(is.null(input$action)){
      
    } else if(input$action == presses[1] & is.null(df$three)) { 
           
        df$three = input$action
     
    } else if(input$action == presses[2]  & is.null(df$eight)){
      
      df$eight = input$action - presses[1]
      
    } else if(input$action == presses[3] & is.null(df$thirteen)) {
      
      df$thirteen = input$action - presses[2]
 }
    
  })
  
  output$countOutput <- renderPrint({
    
    paste(df$three[1], df$eight[1], df$thirteen[1])
    
  })
  
  
}

shinyApp(ui, server)