
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
Let $a$, $b$, $c$ be three different lines in space, and $\alpha$, $\beta$ be two non-coincident planes in space. Among the following propositions, the inverse propositions that hold are:

①. When $b \subset \alpha$, and $c$ is the projection of $a$ within $\alpha$, if $b \perp c$, then $a \perp b$.  
②. When $b \subset \alpha$, and $c \nsubseteq \alpha$, if $c \parallel \alpha$, then $b \parallel c$.  
③. When $b \subset \alpha$, if $b \perp \beta$, then $\alpha \perp \beta$.  
④. When $c \perp \alpha$, if $c \perp \beta$, then $\alpha \parallel \beta$.

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