How should I make series of API calls with the middle one needing time?

I need to make a series of calls to an API. Here is what should happen:

Call 1: Make the first call which returns an ID.
Call 2: Use the ID to make the second call. The second call will come back with an object which includes a status of either 'P', 'C', or 'E' (Processing, Complete, or Error). Once the second call comes back with 'C', make the third call (or stop the process with an error)
Call 3: Use the ID again to get final processing details.

This is what I have so far, but it does not work and does not include the third call.

this.firstCall(formData)
  .pipe(
    interval(3000)
    .pipe(
      switchMap(res => {
        this.secondCall(res)
          .pipe(
            catchError(error => {
              return of(null);
            })
        )
      }),
      takeUntil(res => res.code !== 'P'),
      timeout(21000)
    )
  )
  .subscribe({
    next: (data) => {
      console.log('SUCCESS DATA', data)
    },
    error: (error) => {
      console.log('PROCESSING ERROR', error)
    }
  })

Solution

I like the elegance of the accepted answer, but for completeness sake, here's a version that doesn't require recursion which can be difficult to grok.

import { of, first, repeat, mergeMap, switchMap, defer } from 'rxjs';

let i = 0;
const firstAPI$ = defer(() => of('123'));
const secondAPI = (id: any) => defer(() => {
  console.log('second API called');
  i++;
  return i > 10 ? of({ code: 'C' }) : of({ code: 'P' });
});
const thirdAPI = (id: any) => of(`hello ${id}`);

firstAPI$
  .pipe(
    switchMap((id: any) =>
      secondAPI(id).pipe(
        // Repeat the subscription to the second API every 500 millis
        repeat({delay: 500}),
        // Take the first non-P value
        first(({code}) => code !== 'P'),
        // Throw the error if the code is 'E' else run the third API
        mergeMap(val => {
          if (val.code === 'E') 
            throw new Error('Processing failed')
          else 
            return thirdAPI(id)
        })
      )
    )
  )
  .subscribe({
    next: (data: any) => {
      console.log('SUCCESS DATA', data);
    },
    error: (error: any) => {
      console.log('PROCESSING ERROR', error);
    },
  });