
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
Triangle \( A B C \) has sides \( AB=1 \), \( BC=\sqrt{7} \), and \( CA=\sqrt{3} \). Let \(\ell_1\) be the line through \( A \) perpendicular to \( AB \), \(\ell_2\) the line through \( B \) perpendicular to \( AC \), and \( P \) the point of intersection of \(\ell_1\) and \(\ell_2\). Find \( PC \).

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