
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
Read the following dialogue and then answer the questions:<br/>Teacher: We define a new type of triangle, a triangle in which the sum of the squares of two sides is equal to twice the square of the third side is called a peculiar triangle.<br/>Xiao Hua: An equilateral triangle must be a peculiar triangle$!$<br/>Xiao Ming: Then, does a right triangle exist as a peculiar triangle?<br/>$(1)$ According to the definition of "peculiar triangle," determine whether Xiao Hua's proposition, "An equilateral triangle must be a peculiar triangle," is true or false.<br/>$(2)$ In $Rt\triangle ABC$, with side lengths $a=5\sqrt{2}$ and $c=10$, is this triangle a peculiar triangle? Please explain your reasoning.<br/>$(3)$ In $Rt\triangle ABC$, with $\angle C=90^{\circ}$, $AB=c$, $AC=b$, $BC=a$, and $b \gt a$, if $Rt\triangle ABC$ is a peculiar triangle, find the values of $a:b:c$.

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