
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 $P$ be the point on line segment $\overline{AB}$ such that $AP:PB = 4:1.$ Suppose further that $\overrightarrow{A} = (1,2)$ and $\overrightarrow{B} = (4,3)$. Then, express $\overrightarrow{P}$ as a combination of $\overrightarrow{A}$ and $\overrightarrow{B}$, and determine the ordered pair $(t, u)$ such that
\[\overrightarrow{P} = t\overrightarrow{A} + u\overrightarrow{B}.\]

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