
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

Given a convex pentagon \( \mathrm{ABCDE} \) with \( \mathrm{AB}=2 \), \( \mathrm{BC}=2 \), \( \mathrm{CD}=5 \), and \( \mathrm{DE}=7 \). It is known that \( \mathrm{AC} \) is the largest side in triangle \( \mathrm{ABC} \) (i.e., sides \( \mathrm{AB} \) and \( \mathrm{BC} \) are strictly less than \( \mathrm{AC} \)), and \( \mathrm{CE} \) is the smallest side in triangle \( \mathrm{ECD} \) (i.e., sides \( \mathrm{DE} \) and \( \mathrm{CD} \) are strictly greater than \( \mathrm{CE} \)). It is also known that in triangle \( \mathrm{ACE} \), all sides are integers and have different lengths. Find the possible lengths of side \( \mathrm{AE} \).

If there are multiple possible answers, list them in ascending order, separated by commas or semicolons.

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