
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
Three different lines $a$, $b$, $c$ are known to be in the same plane. The following four propositions are given: ① If $a$ is parallel to $b$ and $a$ is perpendicular to $c$, then $b$ is perpendicular to $c$; ② If $b$ is parallel to $a$ and $c$ is parallel to $a$, then $b$ is perpendicular to $c$; ③ If $b$ is perpendicular to $a$ and $c$ is perpendicular to $a$, then $b$ is perpendicular to $c$; ④ If $b$ is perpendicular to $a$ and $c$ is perpendicular to $a$, then $b$ is parallel to $c$. Determine which propositions are true. (Fill in the numbers of all true propositions)

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