
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 $\ell$ and $m$ be two non-coplanar lines in space, and let $P_{1}$ be a point on $\ell$. Let $P_{2}$ be the point on $m$ closest to $P_{1}$, $P_{3}$ be the point on $\ell$ closest to $P_{2}$, $P_{4}$ be the point on $m$ closest to $P_{3}$, and $P_{5}$ be the point on $\ell$ closest to $P_{4}$. Given that \(P_{1} P_{2}=5\), \(P_{2} P_{3}=3\), and \(P_{3} P_{4}=2\), compute \(P_{4} P_{5}\).

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