Something in the back of my mind for some time, but I haven't really thought it out:

The first thing we suggested was to take a survey.

The first thing we planned was to take a survey.

The first thing we accomplished was to take a survey.

The first thing we intended was to take a survey.

The first thing we did was to take a survey.

The first thing we did was take a survey.

I think these are all correct.

The question is why an infinitive with antecedent 'do' may shed its 'to'.

I can't think of any other antecedent verbs that can abbreviate the infinitive in this construction.

Anyone care to elaborate?
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Just a guess:

It's not the bare infinitive, it's the base form.

The verb "do" can function both as an auxilliary and as a main finite verb.

We take surveys / We do take surveys

He takes surveys / He does take surveys

We took surveys / We did take surveys

When we use an auxilliary, the main verb takes the base form, and the auxilliary becomes inflected.

So this special case for "do" derives from its function as an auxilliary.

It seems to straddle the fence in your examples, being both the main verb and an auxilliary.

What he does is take surveys.

What I did was take surveys.

Of course you can also use the infinitive.

I'm not suggesting that this is a standard grammatical structure.

It's a special case, an anomaly, which derives from our being accustomed to hearing the "do" as an auxilliary.

I'm sure there are articles somewhere which put an authoritative complexion on this.
electrumThe question is why an infinitive with antecedent 'do' may shed its 'to'.
I believe the answer can be traced to the fact that "do" is a 'pro-verb'.

electrumI can't think of any other antecedent verbs that can abbreviate the infinitive in this construction.
Me neither, but then no other verb is a 'pro-verb'.

What did you do? I spoke to him. That's what I did. What I did was ... All I did was ...

What did you do? I ate lunch. That's what I did. The first thing I did was ...

What did you do? I swam for three hours. That's what I did. The only thing I did was ...

CJ
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CalifJimI believe the answer can be traced to the fact that "do" is a 'pro-verb'.
Saints be praised.
Then let us do so with dispatch.

This is what I was hoping for!

Thanks, CJ. Emotion: happy
Bascially, you're right. but it's still an infinitive.

dictionary.com:

(in English) the simple or basic form of the verb, as come, take, eat, be, used after auxiliary verbs, as in I didn't come, He must be, or this simple form preceded by a function word, as to in I want to eat.

I thought of this oddity:

He should do take a survey...Wrong

The first thing he should do is take a survey...Right
electrumbut it's still an infinitive.
I'm not clear on the referent of "it", nor why "its" being an infinitive is relevant.

CJ
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What about?

He dares not take a survey.

What he dares not is take a survey.

Just teasing.
Bascially, you're right. but it's still an infinitive.

dictionary.com:

(in English) the simple or basic form of the verb, as come, take, eat, be, used after auxiliary verbs, as in I didn't come, He must be, or this simple form preceded by a function word, as to in I want to eat.

I thought of this oddity:

He should do take a survey...Wrong

The first thing he should do is take a survey...Right
Oops! That was intended for Avangi.
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