
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  $ABC$  has  $\angle A=90^\circ$ , side  $BC=25$ ,  $AB>AC$ , and area  $150$ .  Circle  $\omega$  is inscribed in  $ABC$ , with  $M$  its point of tangency on  $AC$ .  Line  $BM$  meets  $\omega$  a second time at point  $L$ .  Find the length of segment  $BL$ .

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