
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 $\mathrm{ABCDE}$ where $\mathrm{AB} = 2$, $\mathrm{BC} = 2$, $\mathrm{CD} = 5$, and $\mathrm{DE} = 7$. It is known that $\mathrm{AC}$ is the longest side in triangle $\mathrm{ABC}$ (i.e., sides $\mathrm{AB}$ and $\mathrm{BC}$ are strictly less than $\mathrm{AC}$), and $\mathrm{CE}$ is the shortest side in triangle $\mathrm{ECD}$ (i.e., sides $\mathrm{DE}$ and $\mathrm{CD}$ are strictly greater than $\mathrm{ED}$). Additionally, it is known that in triangle $\mathrm{ACE}$, all sides are integers and have different lengths. Find the possible lengths of side $\mathrm{AE}$.

Write the possible answers in ascending order, separated by commas or semicolons.

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