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Can I say,

(a) Long ago, a rabbit and a tortoise were best / close friends. The rabbit and his friends invited the tortoise to race. It was very happy and promised the rabbit to race. At first / First / Firstly, the rabbit was fast and the tortoise was far behind / far (behind) .

(b) It was a bright and sunny Sunday morning. The rabbit invited the tortoise to race.

When the race began, the rabbit ran fast but the tortoise was slow. / the tortoise crawled slowly (far) behind. After awhile, the rabbit took a rest under a shady tree and felt asleep (soon).
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Vincent Teo(c) Last Sunday, there was a race between a rabbit and a tortoise in the jungle. Many friends were invited to the race. Then, the race began.

OK
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When the rabbit woke up, he was shocked.

He saw the tortoise already reaching the finishing line / saw the tortoise was at the finishing line.>> Either are OK

He ran faster to catch up with the tortoise but he failed and was too late. So the tortoise had won / won (the race).
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Comments  
(c) Last Sunday, there was a race between a rabbit and a tortoise in the jungle. Many friends were invited to the race. Then, the race began.
 Philip's reply was promoted to an answer.
Vincent Teobest / close friends.

either
Vincent Teopromised the rabbit to race.
OK
Vincent TeoAt first
This is better.
Vincent Teowas far behind
this one
Vincent Teotortoise was slow.
slower
Vincent Teofelt asleep (soon).
Soon fell asleep
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Can I say,

When the rabbit woke up, he was shock. He saw the tortoise already reaching the finishing line / saw the tortoise was at the finishing line. He ran faster to chase the tortoise but he failed and was late. So the tortoise had won / won (the race).
 LouiseT's reply was promoted to an answer.