
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
The base of the pyramid $A B C E H$ is a convex quadrilateral $A B C E$, which is divided into two congruent triangles by the diagonal $B E$. The length of edge $A B$ is 1, the lengths of edges $B C$ and $C E$ are equal. The sum of the lengths of edges $A H$ and $E H$ is $\sqrt{2}$. The volume of the pyramid is $1 / 6$. Find the radius of the sphere with the maximum volume that can fit inside the pyramid.

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