Empezar, comenzar, iniciar

I was practicing and wanted to translate the sentence "We can start to walk." and the correct answers were these:
1. Podemos empezar caminando.
2. Podemos comenzar a caminar.

But I don't understand why there weren't answers like:
"Podemos empezar a caminar." or "Podemos comenzar caminando." ?
Are they correct, too?

And please, can someone explain when to use these words: empezar, comenzar, iniciar?

NOTA: Don't worried about that, the means of them are very similarr, INICIAR is a bit more cultured and used to politas issues, sports......

empezar, comenzar, iniciar

I was practicing and wanted to translate the sentence "We can start to walk." and the correct answers were these:
1. Podemos empezar caminando.
2. Podemos comenzar a caminar.

But I don't understand why there weren't answers like:
"Podemos empezar a caminar." or "Podemos comenzar caminando." ?
Are they correct, too?

And please, can someone explain when to use these words: empezar, comenzar, iniciar?

Los verbos comenzar y empezar significan lo mismo. En España preferimos el verbo empezar y en América Latina el verbo comenzar, pero se usa habitualmente los dos. En cuanto al verbo iniciar, el siginificado es el mismo, pero se usa en términos más cultos. Por ejemplo: Los dos gobiernios han iniciado las negociaciones.

Espero haberte ayudado.