
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given that $A$ represents a point, $a$, $b$, $c$ represent lines, $M$, $N$ represent planes, the following propositions are given:
   \[①a⊥M,\] if \[M⊥N,\] then \[a/\\!/N\]       
   \[②a⊥M,\] if \[b/\\!/M,\] \[c/\\!/a,\] then \[a⊥b,\] \[c⊥b\]
   \[③a⊥M,\] \[b⊄M,\] if \[b/\\!/M,\] then \[b⊥a\]
   \[④a⊂β,\] \[b∩β=A,\] \[c\] is the projection of \[b\] on \[β,\] if \[a⊥c,\] then \[a⊥b.\]

Among these propositions, the correct one(s) is/are \_\_\_\_\_\_.

Present the answer in LaTex format: \boxed{Your answer}
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