
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 triangle $\triangle ABC$, $AB=4$, $AC=3$, $\angle BAC=90^{\circ}$. Point $D$ lies on segment $BC$ (point $D$ does not coincide with endpoints $B$ and $C$), extend $AD$ to $P$ such that $AP=9$, $\overrightarrow{PA}=m\overrightarrow{PB}+({\frac{3}{2}-m})\overrightarrow{PC}$ (where $m$ is a constant).<br/>($i$) If $\overrightarrow{PA}=\lambda\overrightarrow{PD}$, then $\lambda =$______;<br/>($ii$) The length of segment $CD$ is ______.

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