
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
A lake is shaped like a trapezoid with parallel sides measuring $200 \mathrm{~m}$ and $100 \mathrm{~m}$. The angles of the other two sides with these parallel sides are $90^{\circ}$ and $45^{\circ}$. Two guards walk around the lake in the same direction and at the same pace, maintaining an equal distance from each other along the perimeter of the lake. What is the maximum straight-line distance between the two guards?

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