
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
On a birthday card printed with April 29, a child placed two positive integers \(x\) and \(y\) between 4 and 29, resulting in a five-digit number \(\overline{4 x y 29}\). It turns out that this number is exactly the square of a positive integer \(T\) corresponding to the child's birthday: \(\overline{4 x y 29} = T^2\). What is the positive integer \(T\) corresponding to the child's birthday?

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