
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 two distinct lines $a$ and $b$, and two distinct planes $α$ and $β$, evaluate the following propositions:

$①$ If $a$ is parallel to $α$ and $b$ is parallel to $α$, then $a$ is parallel to $b$;

$②$ If $a$ is perpendicular to $α$ and $a$ is perpendicular to $β$, then $α$ is parallel to $β$;

$③$ If $α$ is perpendicular to $β$, then there exists a plane $γ$ such that $γ$ is perpendicular to both $α$ and $β$;

$④$ If $α$ is perpendicular to $β$, then there exists a line $l$ such that $l$ is perpendicular to $α$ and $l$ is parallel to $β$.

Identify the sequence numbers of all true propositions above.

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