
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
Let  $A_1$  be a midmoint of  $BC$ , and  $G$  is a centroid of the non-isosceles triangle  $\triangle ABC$ .  $GBKL$  and  $GCMN$  are the squares lying on the left with respect to rays  $GB$  and  $GC$  respectively. Let  $A_2$  be a midpoint of a segment connecting the centers of the squares  $GBKL$  and  $GCMN$ . Circumcircle of triangle  $\triangle A_{1}A_{2}G$  intersects  $BC$  at points  $A_1$  and  $X$ . Find  $\frac{A_{1}X}{XH}$ , where  $H$  is a base of altitude  $AH$  of the triangle  $\triangle ABC$ .

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