
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  $ABC$ be an isosceles triangle with  $\left|AB\right| \equal{} \left|AC\right| \equal{} 10$  and  $\left|BC\right| \equal{} 12$.  $P$ and $R$ are points on $\left[BC\right]$ such that $\left|BP\right| \equal{} \left|RC\right| \equal{} 3$.  $S$ and $T$ are midpoints of $\left[AB\right]$ and $\left[AC\right]$, respectively. If $M$ and $N$ are the foot of perpendiculars from $S$ and $R$ to $PT$, then find $\left|MN\right|$.

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