
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
Let triangle \( ABC \) be such that \( AB = AC = 22 \) and \( BC = 11 \). Point \( D \) is chosen in the interior of the triangle such that \( AD = 19 \) and \( \angle ABD + \angle ACD = 90^\circ \). The value of \( BD^2 + CD^2 \) 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