
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
In a convex pentagon  $ABCDE$  the sides have lengths  $1,2,3,4,$  and  $5$ , though not necessarily in that order.  Let  $F,G,H,$  and  $I$  be the midpoints of the sides  $AB$ ,  $BC$ ,  $CD$ , and  $DE$ , respectively.  Let  $X$  be the midpoint of segment  $FH$ , and  $Y$  be the midpoint of segment  $GI$ .  The length of segment  $XY$  is an integer.  Find all possible values for the length of side  $AE$ .

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