
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 the quadrilateral $ABCD$ in the plane, $\angle ABC=90^{\circ}$, $\angle C=135^{\circ}$, $BD=\sqrt{5}$, $CD=\sqrt{2}$.  
$(1)$ Find $\cos \angle CBD$;  
$(2)$ If $\triangle ABD$ is an acute triangle, find the range of values for the area of $\triangle ABD$.

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