
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 a triangle  $ABC$  such that  $\angle{BAC}=90^\circ, \ |\overrightarrow{AB}|=1,\ |\overrightarrow{AC}|=\sqrt{3}$ ,  a point  $P$  inside the  $\triangle{ABC}$  satisfies  $\frac{\overrightarrow{PA}}{|\overrightarrow{PA}|}+\frac{\overrightarrow{PB}}{|\overrightarrow{PB}|}+\frac{\overrightarrow{PC}}{|\overrightarrow{PC}|}=\overrightarrow{0}.$ (1) Find  $\angle{APB},\ \angle{APC}.$ (2) Find  $|\overrightarrow{PA}|,\ |\overrightarrow{PB}|,\ |\overrightarrow{PC}|.$ 

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