Either "himself" or "herself" is most accepted by prescriptive grammarians.
"Herself" is increasingly used by itself in academic writing to counterbalance the sexist implications of assuming that everyone is male (by using the male form as the generic for all people).
"Themselves" is probably most common but (practically speaking) could be considered more casual.
I do not believe I have ever seen "themself" and would personally consider it the least appealing of all possible options.
A native speaker might use it to make a point or promote language change, but in an obviously non-native speaker it would almost certainly be taken as an error.
If I were you, I would go with "him or herself" or "themselves."
ell.stackexchange.com/users/8389/Tiercelet
Almost all English speakers and writers use "themselves."
This question was studied in 2009 by Collins Dictionaries for the Committee on Bible Translation.
The conclusion is that the plural generic is almost universal and increasing, against masculine generic and other alternatives.
http://www.niv-cbt.org/information/collins-corpus-report/ 1.
Generic pronouns and determiners
This part of the study considered the types of pronouns and determiners that are used to refer to indefinite pronouns (such as someone, everybody and one) and non-gender specific nouns (such as a person, each child and any teacher): A. masculine (he, his, himself, etc.);
B. feminine (she, her, herself, etc.); C. plural/gender-neutral (they, them, one, themselves, etc.); D. alternative forms (s/he, him or her, his/her, etc.)
In all the varieties of English analyzed, plural/neutral pronouns and determiners account for the majority of usages.
Between 1990 and 2009, instances of masculine generic pronouns and determiners, expressed as a percentage of total generic pronoun usage in general written English, fell from 22% to 8%.
e.g. ‘…when a person accepts unconditional responsibility, he denies himself the privilege of “complaining” and “finding faults.”’ Instances of ‘alternative’ generic pronouns and determiners fell from 12% to 8%.
e.g. ‘Any citizen who wants to educate himself or herself has plenty of sources from which to do so.’
Instances of plural/neutral generic pronouns and determiners rose from 65% to 84%.
e.g. ‘If you can identify an individual who metabolises nicotine faster you can treat them more effectively.’
Figures for the other corpora analyzed in the study are broadly comparable with figures from the general written English corpus both in overall magnitude and in the general trend over time.
ell.stackexchange.com/users/8519/Dave Burt
There is no universally accepted "correct" answer: whatever you choose, himself or herself or themselves or themself , you are going to annoy some of your readers.
If you care, the only solution is to find another way of saying it.
Anyone can find the answer.
Anyone can find the answer without help.
You can find the answer yourself (and
so can anyone else).
The answer is left to the reader to find.
—and so forth.
ell.stackexchange.com/users/32/StoneyB on hiatus
I would go with "themself."
Singular
they/them has a long history in the English language, and has come back into style with a lot of mainstream publications as a way to avoid inserting gender into general statements like the one you're making.
ell.stackexchange.com/users/3464/wordsmythe
"Himself" or "herself" are both correct.
"Themselves" is technically incorrect, but has found common usage lately due to concerns about sexism.
The most practical advice is to avoid the idiom, e.g. with something like
"Anyone can find the answer without help."
(slightly changes the meaning)
"One can find the answer oneself.
" (rather formal) "One can find the answer on one's own."
(still kind of formal)
"Anyone can find the answer using only common sense."
(getting casual and changes the meaning somewhat)
ell.stackexchange.com/users/8511/John Wu
I am wondering which following form is correct:   Anyone can find the answer himself.  Anyone can find the answer themselves.   More generally, what reflexive pronoun should I use with anyone/anybody?
Anyone can find the answer himself vs. anyone can find the answer themselves: what reflexive pronoun should I use with anyone/anybody?
ell.stackexchange.com
<genderless-pronouns><reflexive-pronouns><singular-they>
ell.stackexchange.com/questions/27324
ell.stackexchange.com/users/3023/Franck Dernoncourt
There is no universally accepted "correct" answer. Himself,  herself, themself or  themselves are all valid. "Themselves" is probably most common and technically incorrect -  it could be considered more casual but has found common usage lately due to concerns about sexism. Himself/herself/themself are all correct. While technicallyincorrect, themselves is widely used.
Himself/herself/themself are all correct. Themselves is widely used but technically incorrect.