
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 the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$ respectively. Given that $\sin B - \sin C = \sin (A-C)$.  
$(1)$ Find the value of $A$.  
$(2)$ If $AB=2$, $AC=5$, and the medians $AM$ and $BN$ on sides $BC$ and $AC$ intersect at point $P$, find the cosine value of $\angle MPN$.

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