
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

On the sides $AB, BC, AC$ of an equilateral triangle $ABC$ respectively, points $A_1, B_1, C_1$ are placed such that $A A_1 = B B_1 = C C_1 = x$. The side length of the triangle is $a$. Find such an $x$ that the ratio of the areas of triangles $A_1 B_1 C_1$ and $ABC$ is equal to $m$. Within what range can the value of $m$ vary?

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