Very difficult to reach truss-rod with provided Allen key

Question

Disclaimer. I am a total guitar beginner; I have no intention of messing with the truss rod if there is no need to, but after I bought my strat-type, I started noticing what I think are intonation problems and decided to spend some time studying the anatomy of the instrument to see where the problem may be. I don't think that the real problem is the neck relief here, but what I noticed on the neck seems weird to me and opinions are appreciated.

This is a picture of my strat-type guitar with an Allen Key fully insterted into the truss rod nut. This is not any key; it is actually the one that came with the instrument.

I am perplexed because the wrench is very close to the wood. Even if the strings were not there, I would not be able to turn the key without damaging the sides of the truss rod hole.

Actually, I tried loosening the strings, removing the string trees, and tried to use the long leg of the Allen key (like this guy here, see the 7th picture). Well, if I do this, I am not even able to reach the nut. I believe the problem in this case is the lower side of the truss rod hole, that is preventing me from aligning the key to the nut.

In short, it seems that the geometry of that thing is a little messed up. As I said before, that key came with the instrument, and its short leg is actually longer than other Allen keys that I have in my house; this means that with those other Allen keys I would not even reach the nut.

• Is there something wrong with the neck?
• Is the truss rod broken or went out of place?
• What does the truss rod nut look like on real Strats?
• Would I need something like this if I were do adjust relief?

More info

The guitar is a Glarry GST.

Update

From the answers and comments, I gather that it is advisable to have a special tool. This and this look like they are the same thing, with the non-fender one being much cheaper.

Update #2

@Tetsujin is right about sizes. In the end I got a 4 mm Allen T-key with ball end from the same manufacturer, which is the metric version (i.e. it's basically a little thinner) of the imperial one that I posted before. I can confirm that the imperial one will not fit the nut on my Glarry. If you have a Fender or Squier I presume you should be fine with the imperial one, but I can't say for sure.

Answer 1

The truss rod on real Strats is at the other end. You can't even see it until you take the scratch plate off, or reach it until you loosen the neck. You got an easy one there ;)

and yes, get a long key with a ball-end, like the Fender one. All you're fighting with after that is they put an 'overkill' string-guide in the way.

The long handle means you're not fighting against the head or access hole edges, the ball-end gives you flexibility as it doesn't have to be in the socket dead straight, it works at an angle.

Blow-up of linked advert, showing ball-end

I suppose I should mention that a Strat copy will not necessarily use the same American allen key sizes as Fender, they're quite likely to be metric, so double check before you buy.