
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
Suppose plane $α$ intersects plane $β$ at line $EF$, with $AB \perp α$ and $CD \perp β$, the feet of the perpendiculars being $B$ and $D$ respectively. If another condition is added, it can be deduced that $BD \perp EF$. Now, we have the following four conditions:

1. $AC \perp α$;
2. The angle between $AC$ and plane $α$ is equal to the angle between $AC$ and plane $β$;
3. The projection of $AC$ onto plane $β$ lies on the same line as $BD$;
4. $AC \parallel EF$.

Which of the above conditions could be added to help deduce that $BD \perp EF$? Fill in the correct condition number(s).

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