
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
$(1)$ Given points $M(3,2)$ and $N(4,-1)$, where point $P$ lies on the $x$-axis and $\angle MPN$ is a right angle, find the coordinates of point $P$.  
$(2)$ Given points $A(7,-4)$ and $B(-5,6)$, find the equation of the perpendicular bisector of segment $AB$.

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