
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 necessary and sufficient condition for two triangles with side lengths $a, b, c$ and side lengths $a_{1}, b_{1}, c_{1}$ to be similar is:
$$
\sqrt{a a_{1}}+\sqrt{b b_{1}}+\sqrt{c c_{1}}=\sqrt{(a+b+c)\left(a_{1}+b_{1}+c_{1}\right)} .
$$

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