
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
Given line $l_{1}: 2x-y+1=0$, line $l_{2}: 4x-2y+a=0$, and circle $C: x^{2}+y^{2}-2x=0$, if the sum of the distances from any point $P$ on circle $C$ to the two lines $l_{1}$ and $l_{2}$ is a constant $2\sqrt{5}$, then the real number $a=$ \_\_\_\_\_\_.

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