
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
In the right triangle $ABC$, point $M$ moves along the leg $CA$ from $A$ towards $C$. The line passing through $M$ and perpendicular to $BM$ intersects the hypotenuse $AB$ at point $N$. The projection of $N$ onto $CA$ is point $P$. How close can $P$ get to $C$? Construct this closest position.

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