
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
Parallel lines \(\ell_1, \ell_2, \ell_3, \ell_4\) are evenly spaced in the plane, in that order. Square \(ABCD\) has the property that \(A\) lies on \(\ell_1\) and \(C\) lies on \(\ell_4\). Let \(P\) be a uniformly random point in the interior of \(ABCD\) and let \(Q\) be a uniformly random point on the perimeter of \(ABCD\). Given that the probability that \(P\) lies between \(\ell_2\) and \(\ell_3\) is \(\frac{53}{100}\), the probability that \(Q\) lies between \(\ell_2\) and \(\ell_3\) can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a + b\).

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