
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 \(ABCD\) be a rhombus with \(AC = 20\) and \(BD = 40\). Let \(M\) be a point on \(\overline{BD}\), exactly \(10\) units from \(B\). Let \(R\) and \(S\) be the feet of the perpendiculars from \(M\) to \(\overline{AC}\) and \(\overline{AB}\), respectively. Compute the length of \(RS\).
A) \( 5\sqrt{2} \)
   B) \( 10\sqrt{2} \)
   C) \( 15\sqrt{2} \)
   D) \( 20\sqrt{2} \)

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