
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 the hyperbola $C:\frac{x^2}{a^2}-y^2=1\;(a>0)$ intersect the line $l:x+y=1$ at two distinct points $A$ and $B$.  
$(1)$ Find the range of real numbers for $a$.  
$(2)$ If the intersection point of the line $l$ and the $y$-axis is $P$, and $\overrightarrow{PA}=\frac{5}{12}\overrightarrow{PB}$, find the value of the real number $a$.

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