Options 3 & 4 both mean "must not" because they are both variations on べきではない tested at JPLT N1 (in fact べからざる is a variation on べからず (see on)). The trick is to understand which best fits the context of the sentence when we apply "usual" JLPT level N1 definitions but even if we do not fully understand the context we can still get the question right if we know the grammatical rules (接続の仕方).
べからず is used for injunctions (ie prohibitions such as "Do not..") and is an old written form not generally used but sometimes written on bulletin board/notices. As a result it is quite likely to appear in a quotation. eg:
(公演で）「芝生に入るべからず」｜([Notice] at the park) "Keep off the grass"
べからざる is the form of べからず that modifies nouns ie VるべからざるN. eg:
彼は我がチームには欠くべからざる選手である。｜Mr A is (an) indispensable (player) on the team.
(And, just for the sake of completeness, べき is the form of べく that modifies nouns ie VるべきN.]
APPLICATION (COMPREHENSIVE ANSWER)
If we apply these principles to the question:
1.べく 2.べき 3.べからざる 4. べからず
Step 1: Read and (try to*) understand the context of the passage to eliminate the obvious incorrect answers:
A: "Although it says _ be distracted by the rumours,...."
B: "I just took them as face value, didn't I."
The gap is most likely to be filled by a prohibition (ie an injunction such as "Do not...") rather than a command ("Do...."). This eliminates options 1 & 2.
Step 2: Look at next option: 3. べからず is used for injunctions and this is a quotation so without looking at 4 we know this is a good candidate for the correct answer.
Step 3: Option 3 is confirmed as correct by looking at the remaining option: 4 べからざる is just a variation on 3. べからず which is used to modify nouns (VるべからざるN). Although it literally means the same as 3 grammatically it does not fit as well / at JLPT level N1.
*Note: I am sure somebody can improve on my translation but it is good enough for the question!
References: The references I used to write this answer were 日本語表現文型辞典 ＆ 総まとめ 文法N1 but once you've done enough questions you can intuitively "guess" the right answer just using step one (that's what I did here and last July).