
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
The triangle  $ABC$  has a right angle at  $C.$  The point  $P$  is located on segment  $AC$  such that triangles  $PBA$  and  $PBC$  have congruent inscribed circles. Express the length  $x = PC$  in terms of  $a = BC, b = CA$  and  $c = AB.$ 

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