Answer:

To understand why the correct answer is "have written," let's break down the sentence and analyze the verb tense and usage.

Past Progressive:
The verb tense used in the sentence is the past tense, which indicates that the action took place in the past. However, the past progressive tense (e.g., "was writing") is not suitable in this context because it implies an ongoing or incomplete action. Since the letter has been completed, the past progressive tense is not appropriate.

Present Perfect:
The present perfect tense (e.g., "have written") is used to describe an action that was completed in the past but has a connection to the present. In this case, the action of writing the letter is finished, but it is still relevant because the speaker cannot find a stamp to send it. The present perfect tense suggests that the action has an impact on the present moment.

Key Pointers:
- The sentence is in the past tense.
- The past progressive tense (e.g., "was writing") does not fit because the action is completed.
- The present perfect tense (e.g., "have written") is used to describe a completed action with a connection to the present.
- The action of writing the letter is finished, but the speaker cannot find a stamp to send it, which is relevant in the present.

Conclusion:
Considering the above points, the correct answer is "have written." It accurately reflects that the action of writing the letter is completed, but the inability to find a stamp creates a connection to the present moment. Thus, the present perfect tense is the most suitable option in this context.