
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 $\overline{AC}$ and $\overline{BC}$ be line segments with $P$ the point on segment $\overline{AC}$ such that $AP:PC = 4:1$. The point $Q$ on segment $\overline{BC}$ satisfies $BQ:QC = 1:4$. Find the constant $r$ and $s$ if $P$ and $Q$ can be expressed as weighted averages of $A$, and $C$, and $B$, and $C$ respectively:
\[
\overrightarrow{P} = r\overrightarrow{A} + (1-r)\overrightarrow{C} \quad \text{and} \quad \overrightarrow{Q} = s\overrightarrow{B} + (1-s)\overrightarrow{C}.
\]

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