
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  $A$ ,  $B$ ,  $C$ ,  $D$ ,  $E$ ,  $F$ ,  $G$ ,  $H$ ,  $I$  be nine points in space such that  $ABCDE$ ,  $ABFGH$ , and  $GFCDI$  are each regular pentagons with side length  $1$ . Determine the lengths of the sides of triangle  $EHI$ .

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