How many days before Expiry of H1 i can enter in US?

I have H1 Visa (3 years) which is getting expired in 1 month. I did not travel on this visa till now. My employer is now giving me new offer and saying extension will be filled as soon as i enter in USA. I can stay on that extension till the decision comes.

My Question is can i enter on H1 which is going to expire in 1 month or less of my travel date in US?

Can i fill extension before i enter USA?

Please replay ASAP.

Thanks

This is unclear and depends on the CBP officer. But its probably better to just go ahead and file in premium and enter. I only take a few days to get the receipt, that proves that you have filed an extension.