
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  $\ell$  be a line and let points  $A$ ,  $B$ ,  $C$  lie on  $\ell$  so that  $AB = 7$  and  $BC = 5$ . Let  $m$  be the line through  $A$  perpendicular to  $\ell$ . Let  $P$  lie on  $m$ . Compute the smallest possible value of  $PB + PC$ . 

*Proposed by Ankan Bhattacharya and Brandon Wang*

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