
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
In the plane rectangular coordinate system \( x O y \), three points \( A(3,5), B(1,3), C(4,3) \). Translate \(\triangle ABC\) to the right-bottom to the new position \(\triangle A_1 B_1 C_1\), where the direction of translation forms an angle of \(30^{\circ}\) with the \(x\)-axis. If the area of the common part of \(\triangle ABC\) and \(\triangle A_1 B_1 C_1\) is equal to \(m\) times the area of \(\triangle ABC\) (\(0 < m < 1\)), then the translation distance \(d\) is equal to _______.

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