
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 \( A, B, C, D \) be points chosen on a circle, in that order. Line \( BD \) is reflected over lines \( AB \) and \( DA \) to obtain lines \( \ell_1 \) and \( \ell_2 \) respectively. If lines \( \ell_1 \), \( \ell_2 \), and \( AC \) meet at a common point and if \( AB = 4 \), \( BC = 3 \), \( CD = 2 \), compute the length \( DA \).

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