
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 vertex of a cone is $S$, the cosine value of the angle formed by the generatrix $SA$ and $SB$ is $\frac{7}{8}$, and the angle between $SA$ and the base of the cone is $45^{\circ}$. If the area of $\triangle SAB$ is $5\sqrt{15}$, then the lateral area of the cone is ____.

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