
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 convex quadrilateral \( ABCI \), \( AB = a \), \( BC = b \), \( CI = c \), \( DA = d \), \( AC = e \), \( BD = f \), and \( H = \max \{a, b, c, d, e, f\} = 1 \). Find the maximum value of \( abcd \).

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