
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
$\triangle ABC$ is an isosceles triangle in which $AB = AC = 10$ and $BC = 12$. The points $S$ and $R$ are on $BC$ such that $BS: SR: RC = 1: 2: 1$. The midpoints of $AB$ and $AC$ are $P$ and $Q$ respectively. Perpendiculars are drawn from $P$ and $R$ to $SQ$ meeting at $M$ and $N$ respectively. Calculate the length of $MN$.

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