
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
Given that the focus of the parabola $C:y^{2}=4x$ is $F$, and the line $l$ with a slope of $1$ intersects $C$ at points $A$ and $B$ in the first and fourth quadrants respectively, with point $P$ on the $x$-axis. <br/>$(1)$ When $|AF|+|BF|=10$, find the coordinates of point $P$; <br/>$(2)$ Let $\overrightarrow{AP}=λ\overrightarrow{PB}$, if $|AB|=12\sqrt{2}$, find the value of $\lambda$.

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