
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
Squares \(ABCD\) and \(DEFG\) have side lengths 1 and \(\frac{1}{3}\), respectively, where \(E\) is on \(\overline{CD}\) and points \(A, D, G\) lie on a line in that order. Line \(CF\) meets line \(AG\) at \(X\). The length \(AX\) can be written as \(\frac{m}{n}\), where \(m, n\) are positive integers and \(\operatorname{gcd}(m, n)=1\). Find \(100m+n\).

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