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Add time to countdown timer during while counting down in Flutter

I am trying to create a countdown timer application where I can press a button to add more time while the timer is actively counting down just like every microwave oven where there is a button you can press to add an extra minute to it's run time while it's running without stopping anything.

import 'package:flutter/material.dart';

void main() {
  runApp(new MaterialApp(
    home: new MyApp(),
  ));
}

class Countdown extends AnimatedWidget {
  Countdown({ Key key, this.animation }) : super(key: key, listenable: animation);
  Animation<int> animation;

  @override
  build(BuildContext context){
    return new Text(
      animation.value.toString(),
      style: new TextStyle(fontSize: 150.0),
    );
  }
}

class MyApp extends StatefulWidget {
  State createState() => new _MyAppState();
}

class _MyAppState extends State<MyApp> with TickerProviderStateMixin {
  AnimationController _controller;

  static const int kStartValue = 4;

  @override
  void initState() {
    super.initState();
    _controller = new AnimationController(
      vsync: this,
      duration: new Duration(seconds: kStartValue),
    );
  }

  @override
  Widget build(BuildContext context) {
    return new Scaffold(
      floatingActionButton: new FloatingActionButton(
        child: new Icon(Icons.play_arrow),
        onPressed: () => _controller.forward(from: 0.0),
      ),
      body: new Container(
        child: new Center(
          child: new Countdown(
            animation: new StepTween(
              begin: kStartValue,
              end: 0,
            ).animate(_controller),
          ),
        ),
      ),
    );
  }
}

This example from a similar question about timers makes sense to me and has been my jumping off point. I know I need to change the duration and replace the animation with a new one with the appropriate duration, but I never get anything close to the correct behavior I am looking for.

I think the below is what you're trying to accomplish. Tapping the FloatingActionButton adds 5 seconds.

import 'dart:async';
import 'package:flutter/material.dart';

void main() => runApp(MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Countdown Demo',
      theme: ThemeData(
        primarySwatch: Colors.blue,
      ),
      home: MyHomePage(title: 'Flutter Countdown Demo'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  MyHomePage({Key key, this.title}) : super(key: key);

  final String title;

  @override
  _MyHomePageState createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> {
  int _timeRemaining = 10;
  Timer _timer;

  @override
  void initState() {
    _timer = Timer.periodic(Duration(seconds: 1), (Timer t) => _getTime());
    super.initState();
  }

  @override
  void dispose() {
    _timer?.cancel();
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: Text(widget.title),
      ),
      body: Center(
        child: Text('$_timeRemaining'),
      ),
      floatingActionButton: FloatingActionButton(
          child: Icon(Icons.add),
          onPressed: () {
            _timeRemaining += 5;
          }),
    );
  }

  void _getTime() {
    setState(() {
      _timeRemaining == 0 ? _timeRemaining = 0 : _timeRemaining--;
    });
  }
}

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