
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
The base of the right prism $ABC A_{1} B_{1} C_{1}$ is an isosceles triangle $ABC$ with $AB = BC = 5$, $\angle ABC = 2 \arcsin \frac{3}{5}$. A plane perpendicular to the line $A_{1}C$ intersects the edges $AC$ and $A_{1}C_{1}$ at points $D$ and $E$ respectively, with $AD = \frac{1}{3}AC$ and $EC_{1} = \frac{1}{3}A_{1}C_{1}$. Find the area of the cross-section of the prism by this plane.

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