According to the answer key for this question, #4 is the correct response. I believe #1 could be correct too. Would you please help me with this?

........by the complexity of the problem, the team requested more time.

1. Being overwhelmed
2. Having overwhelmed
3. To overwhelm.
4. Overwhelmed

Question

According to the answer key for this question, #4 is the correct response. I believe #1 could be correct too. Would you please help me with this?

........by the complexity of the problem, the team requested more time.

1. Being overwhelmed 
2. Having overwhelmed 
3. To overwhelm.
4. Overwhelmed

Answer

#1 and #4 are equally good.

"Overwhelmed", by itself, acts as an adjective to modify "the team".

"Being overwhelmed" is a clause that can act as an adjective, noun, or adverb.

"The team, being overwhelmed, requested more time"   (adjective)
"The team resisted being overwhelmed, but it finally gave up"  (noun)

I found it very hard to construct a sentence in which "being overwhelmed" is adverbial.  GPT and I had to work together on this challenge for about ten minutes.  I would try, then GPT would tell me why I was wrong. Then GPT would try and I would say why GPT was wrong.  We went through about 5 iterations of this and finally found a sentence in which, we both agreed, "being overwhelmed" is adverbial.  Here is the result of our collaboration:

"The room had a pink picture on the wall.  We painted the room a light green, but then pink being overwhelmed was no longer a pretty color in that room."

Answer

You're right to think carefully about this — here's why:

The correct answer is #4: Overwhelmed.

Let’s break it down carefully:

The sentence needs a past participle (like "overwhelmed") because the team was affected by the complexity of the problem.
In English, when we start a sentence this way, we often use a past participle to show that the subject is experiencing something.
Example: Surprised by the news, she dropped her phone.
So:
"Overwhelmed by the complexity of the problem, the team requested more time."
➔ Perfect. It means the team felt overwhelmed.

Now about your idea (#1 "Being overwhelmed"):

"Being overwhelmed" is grammatically possible, but it sounds heavier, more complicated, and is less natural at the beginning of this type of sentence.
It would be more typical in longer or formal writing, like:
Being overwhelmed by the complexity of the problem, the team decided to postpone the project.
But even then, native speakers usually prefer the simpler past participle form when possible.
Summary:

#4 "Overwhelmed" is the best and most natural answer — clear and simple.
#1 "Being overwhelmed" could be grammatically possible, but it is less natural and heavier. That’s why it’s not the correct choice for this test.

Answer

1 and 4 are both okay to use.

Answer

Both 1# and 4# are grammatically correct according to  My grammar book. Such as my English grammar book is, it is correctly useful.
and , such as I am on learning English, I maybe useful accidentally.